Lines and Slopes ACT Math Geometry Review and Practice

Lines and Slopes ACT Math Geometry Review and Practice SAT / ACT Prep Online Guides and Tips You’ve dealt with the basics of coordinate geometry and points (and if you haven’t already, you may want to take a minute to refresh yourself) and now it’s time to look at the ins and outs of lines and slopes on the coordinate plane. This will be your complete guide to lines and slopeswhat slopes mean, how to find them, and how to solve the many types of slope and line equation questions you’ll see on the ACT. What are Lines and Slopes? If you’ve gone through the guide on coordinate geometry, then you know that coordinate geometry takes place in the space where the $x$-axis and the $y$-axis meet. Any point on this space is given a coordinate point, written as $(x, y)$, that indicates exactly where the point is along each axis. A line (or line segment) is a marker that is completely straight (meaning it has no curvature). It is made up of a series of points and and connects them together. A slope is how we measure the slant/steepness of a line. A slope is found by finding the change in distance along the y axis over the change in distance along the x axis. You have probably heard how to find a slope by finding the "rise over run." This means exactly the same thingchange in $y$ over change in $x$. $${\change \in y}/{\change \in x}$$ Let's look at an example: Say we are given this graph and asked to find the slope of the line. We must see how both the rise and the run change. To do this, we must first mark points along the line to in order to compare them to one another. We can also make life easier on ourselves by marking and comparing integer coordinates (places where the line hits at a corner of $x$ and $y$ measurements.) Now we have marked our coordinate points. We can see that our line hits at exactly: $(-3, 5)$, $(1, 0)$, and $(5, -5)$. In order to find the slope of the line, we can simply trace our points to one another and count. We've highlighted in red the path from one coordinate point to the next. You can see that the slope falls (has a negative "rise") of 5. This means the rise will be -5. The slope also moves positively (to the right) 4. Thus, the run will be +4. This means our slope is: $-{5/4}$ Properties of Slopes A slope can either be positive or negative. A positive slope rises from left to right. A negative slope falls from left to right. A straight line has a slope of zero. It will be defined by one axis only. $x = 3$ $y = 3$ The steeper the line, the larger the slope. The blue line is steepest, with a slope of $3/2$. The red line is shallower, with a slope of $2/5$ Now that we've gone through our definitions, let us take a look at our slope formulas. Line and Slope Formulas Finding the Slope $${y_2 - y_1}/{x_2 - x_1}$$ In order to find the slope of a line that connects two points, you must find the change in the y-values over the change in the x-values. Note: It does not matter which points you assign as $(x_1, y_1)$ and $(x_2, y_2)$, so long as you keep them consistent. Find the slope of the line with coordinates at (-1, 0) and (1, 3). Now, we already know how to count to find our slope, so let us use our equation this time. ${y_2 - y_1}/{x_2 - x_1}$ Let us assign the coordinate (-1, 0) as $(x_1, y_1)$ and (1, 3) as $(x_2, y_2)$. $(3 - 0)/(1 - -1)$ $3/2$ We have found the slope of the line. Now let's demonstrate why the equation still works had we switched which coordinate points were $(x_1, y_1)$ and which were $(x_2, y_2)$. This time, coordinates (-1, 0) will be our $(x_2, y_2)$ and coordinates (1, 3) will be our $(x_1, y_1)$. ${y_2 - y_1}/{x_2 - x_1}$ $(0 - 3)/(-1 - 1)$ ${-3}/{-2}$ $3/2$ As you can see, we get the answer $3/2$ as the slope of our line either way. The Equation of a Line $$y = mx + b$$ This is called the â€Å"equation of a line,† also known as an line written in "slope-intercept form." It tells us exactly how a line is positioned along the x and y axis as well as how steep it is. This is the most important formula you’ll need when it comes to lines and slopes, so let’s break it into its individual parts. $y$ is your $y$-coordinate value for any particular value of $x$. $x$ is your $x$-coordinate value for any particular value of $y$. $m$ is the measure of your slope. $b$ is the $y$-intercept value of your line. This means that it is the value along the $y$-axis that the line hits (remember, a straight line will only hit each axis a maximum of one time). For this line, we can see that the y-intercept is 3. We can also count our slope out or use two sets of coordinate points (for example, $(-3, 1)$ and $(0, 3)$) to find our slope of $2/3$. So when we put that together, we can find the equation of our line at: $y = mx + b$ $y = {2/3}x + 3$ Remember: always re-write any line equations you are given into this form! The test will often try to trip you up by presenting you with a line NOT in proper form and then ask you for the slope or y-intercept. This is to test you on how well you're paying attention and get people who are going too quickly through the test to make a mistake. What is the slope of the line $3x + 12y = 24$? First, let us re-write our problem into proper form: $y = mx + b$ $3x + 12y = 24$ $12y = -3x + 24$ $y = -{3/12}x + 24/12$ $y = -{1/4}x + 2$ The slope of the line is $-{1/4}x$ Now let’s look at a problem that puts both formulas to work. For some real number A, the graph of the line $y=(A+1)x +8$ in the standard $(x,y)$ coordinate plane passes through $(2,6)$. What is the slope of this line? A. -4B. -3C. -1D. 3E. 7 In order to find the slope of a line, we need two sets of coordinates so that we can compare the changes in both $x$ and $y$. We are given one set of coordinates at $(2, 6)$ and we can find the other by using the $y$-intercept. The $b$ in the equation is the y-intercept (in other words, the point at the graph where the line hits the y-axis at $x = 0$). This means that, for the above equation, we also have a set of coordinates at $(0, 8)$. Now, let’s use both sets of coordinates- $(2, 6)$ and $(0, 8)$- to find the slope of the line: ${y_2 - y_1}/{x_2 - x_1}$ $(8 - 6)/(0 - 2)$ $-{2/2}$ $-1$ So the slope of the line is -1. Our final answer is C, -1. (Note: don’t let yourself get tricked into trying to find $A$! It can become instinct when working through a standardized test to try to find the variables, but this question only asked for the slope. Always pay close attention to what is being asked of you.) Perpendicular Lines Two lines that meet at right angles are called â€Å"perpendicular.† Perpendicular lines will always have slopes that are negative reciprocals of one another. This means that you must reverse both the sign of the slope as well as the fraction. For example, if a two lines are perpendicular to one another and one has a slope of 4 (in other words, $4/1$), the other line will have a slope of $-{1/4}$. Parallel Lines Two lines that will never meet (no matter how infinitely long they extend) are said to be parallel. This means that they are continuously equidistant from one another. Parallel lines have the same slope. You can see why this makes sense, since the rise over run will always have to be the same in order to ensure that the lines will never touch. No matter how far they extend, these lines will never intersect. What is the slope of any line parallel to the line $8x+9y=3$ in the standard $(x,y)$ coordinate plane? F. -8G. $-{8/9}$H. $8/3$J. 3K. 8 First, let us re-write our equation into proper slope-intercept equation form. $8x + 9y = 3$ $9y = -8x + 3$ $y = -{8/9} + 1/3$ Now, we can identify our slope as $-{8/9}$. We also know that parallel lines have identical slopes. So all lines parallel to this one will have the slope of $-{8/9}$. Our final answer is G, $-{8/9}$. A...valiant attempt to be parallel. Typical Line and Slope Questions Most line and slope questions on the ACT are quite basic at their core. You’ll generally see two to three questions on slopes per test and almost all of them will simply ask you to find the slope of a line when given coordinate points or intercepts. The test may attempt to complicate the question by using other shapes or figures, but the questions always boil down to these simple concepts. Just remember to re-write any given equations into the proper slope-intercept form and keep in mind your rules for finding slopes (as well as your rules for parallel or perpendicular lines), and you’ll be able to solve these types of problems easily. What is the slope of the line through $(5,-2)$ and $(6,7)$ in the standard $(x,y)$ coordinate plane? F. $9$G. $5$H. $-5$J. $5/11$K. $-{5/11}$ We have two sets of coordinates, which is all we need in order to find the slope of the line which connects them. So let us plug these coordinates into our slope equation: ${y_2 - y_1}/{x_2 - x_1}$ $(7 - 2)/(6 - -5)$ $5/11$ Our final answer is J, $5/11$ Despite the fact that we are now working with figures, the principle behind the problem remains the samewe are given a set of coordinate points and we must find their slope. From C to D, we have coordinates (9, 4) and (12, 1). So let us plug these numbers into our slope formula: ${y_2 - y_1}/{x_2 - x_1}$ $(1 - 4)/(12 - 9)$ $-3/3$ $-1$ Our final answer is B, $-1$. As you can see, there is not a lot of variation in ACT question on slopes. So long as you keep track of the coordinates you’ve assigned as $(x_1, y_1)$ and $(x_2, y_2)$, and you make sure to keep track of your negatives and positives, these questions should be fairly straightforward. How to Solve a Line and Slope Problem As you go through your line and slope problems, keep in mind these tips: #1: Always rearrange your equation into $y = mx + b$ If you are given an equation of a line on the test, it will often be in improper form (for example: $10y + 15x = 20$). If you are going too quickly through the test or if you forget to rearrange the given equation into proper slope-intercept form, you will misidentify the slope and/or the y-intercept of the line. So always remember to rearrange your equation into proper form as your first step. $10y + 15x = 20$ = $y = -{3/2}x + 2$ #2: Remember your $\rise/\run$ Our brains are used to doing things "in order," so it can be easy to make a mistake and try to find the change in $x$ before finding the change in $y$. Keep careful track of your variables in order to reduce careless mistakes like this. Remember the mantra of "rise over run" and this will help you always know to find your change in $y$ (vertical distance) over your change in $x$ (horizontal distance). #3: Make your own graph and/or count to find your slope Because the slope is always "rise over run," you can always find the slope with a graph, whether you are provided with one or if you have to make your own. This will help you better visualize the problem and avoid errors. If you forget your formulas (or simply don't want to use them), simply draw your own graph and count how the line rises (or falls). Next, trace its "run." By doing this, you will always find your slope. Now let's put your newfound knowledge to the test! Test Your Knowledge Now that we’ve walked through the typical slope questions you’ll see on the test (and the few basics you’ll need to solve them, let’s look at a few real ACT math examples: 1. 2. Which of the following is the slope of a line parallel to the line $y={2/3}x-4$ in the standard $(x,y)$ coordinate plane? A. $-4$B. $-{3/2}$C. $2$D. $3/2$E. $2/3$ 3. When graphed in the standard $(x,y)$ coordinate plane, the lines $x=-3$ and $y=x-3$ intersect at what point? A. $(0,0)$B. $(0,-3)$C. $(-3,0)$D. $(-3,-3)$E. $(-3,-6)$ Answers: D, E, E Answer Explanations: 1. You can solve this problem in one of two waysby counting directly on the graph, or by solving for the changes in $x$ and $y$ algebraically. Let’s look at both methods. Method 1- Graph Counting The question was generous in that it provided us with a clearly marked graph. We also know that our slope is $-{2/3}$, which means that we must either move down 2 and over 3 to the right, or up 2 and over 3 to the left to keep our movement across a negative slope line consistent. If you use this criteria to count along the graph, you will find that you hit no marked points by counting up 2 and over 3 to the left, but you will hit D when you go down 2 and over 3 to the right. So our final answer is D. Method 2- Algebra Alternatively, you can always use your slope formula to find the missing coordinate points. If we start with our coordinate points of $(2, 5)$ and our slope of $-{2/3}$, we can find our next two coordinate points by counting finding the changes in our $x$ and $y$. Our first coordinate point of $(2, 5)$ has a $y$ value of 5. We know, based on the slope of the line that the change in $y$ is +/- 2. So our next coordinate point must have a $y$ value of either: $5 + 2 = 7$ Or $5 - 2 = 3$ This means we can eliminate answer choices B and C. Now we can do the same for our x-coordinate value. We begin with $(2, 5)$, so our $x$ value is 2. Because the line has a slope of $-{2/3}$, our x-coordinate change at a rate of +/- 3. This means our next x-coordinate values must be either: $2 + 3 = 5$ Or $2 - 3 = -1$ Now, we must put this information together. Because our slope is negative, it means that whatever change one coordinate undergoes, the other coordinate must undergo the opposite. So if we are adding the change in $y$, we must then subtract our change in $x$ (or vice versa). This means that our coordinate points will either be $(5, 3)$ or $(-1, 7)$. (Why? Because 5 comes from adding our change in $x$ and 3 comes from subtracting our change in $y$, and -1 comes from subtracting our change in $x$ and 7 comes from adding our change in $y$.) The only coordinates that match are at D, $(5, 3)$. Our final answer is D. 2. This question is simple so long as we remember that parallel lines have the same slopes and we know how to identify the slope of an equation of a line. Our line is already written in proper slope-intercept form, so we can simply say that the line $y = {2/3}x - 4$ has a slope of $2/3$, which means that any parallel line will also have a slope of $2/3$. Our final answer is E, $2/3$ 3. This question may seem confusing if you’ve never seen anything like it before. It is however, a combination of a simple replacement in addition to coordinate points. We are given that $x = -3$ and $y = x - 3$, so let us replace our $x$ value in the second equation to find a numerical answer for $y$. $y = x - 3$ $y = -3 - 3$ $y = -6$ Which means that the two lines will intersect at $(-3, -6)$. Our final answer is E, $(-3, -6)$. A good test deserves a good break, don't you think? The Take-Aways Though the ACT may present you with slightly different variations on questions about lines and slopes, these types of questions will always boil down to a few key concepts. Once you've gotten the hang of finding slopes, you'll be able to breeze through these questions in no time. Make sure to keep track of your negatives and positives and remember your formulas, and you’ll be able to take on these kinds of questions with greater ease than ever before. What’s Next? Whew! You may know all you need to for ACT coordinate geometry, but there is so much more to learn! Check out our ACT Math tab to see all our individual guides to ACT math topics, including trigonometry, solid geometry, advanced integers, and more. Think you might need a tutor? Take a look at how to find the right math tutor for your needs and budget. Running out of time on ACT math? Check out how to buy yourself more time on ACT math and complete your section on time. Looking to get a perfect score? Our guide to getting a 36 on ACT math will help you iron out those problem areas and set you on the path to perfection. Want to improve your ACT score by 4 points? Check out our best-in-class online ACT prep program. We guarantee your money back if you don't improve your ACT score by 4 points or more. Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math lesson, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next. Check out our 5-day free trial:

3 Tips for Writing Your Williams College Supplement

3 Tips for Writing Your Williams College Supplement SAT / ACT Prep Online Guides and Tips Williams is among the most selective colleges in the country, with an acceptance rate of 15 percent. As part of your Williams application, you’ll need to respond to the Williams writing supplement. In this article, we’ll cover the three questions that make up the Williams writing supplement, offer suggestions for what to write about in your essay, and give you tips for crafting the best essay possible. The Williams Writing Supplement There are three different questions on the Williams writing supplement. You need to respond to one of them as part of your application. 1. At Williams we believe that bringing together students and professors in small groups produces extraordinary academic outcomes. Our distinctive Oxford-style tutorial classes- in which two students are guided by a professor in deep exploration of a single topic- are a prime example. Each week the students take turns developing independent work- an essay, a problem set, a piece of art- and critiquing their partner’s work. Focused on close reading, writing and oral defense of ideas, more than 60 tutorials a year are offered across the curriculum, with titles like Aesthetic Outrage, Financial Crises: Causes and Cures, and Genome Sciences: At the Cutting Edge. Imagine yourself in a tutorial at Williams. Of anyone in the world, whom would you choose to be your partner in the class, and why? 2. Each Sunday night, in a tradition called Storytime, students, faculty and staff gather to hear a fellow community member relate a brief story from their life (and to munch on the storyteller’s favorite homemade cookies). What story would you share? What lessons have you drawn from that story, and how would those lessons inform your time at Williams? 3. Every first-year student at Williams lives in an Entry- a thoughtfully constructed microcosm of the student community that’s a defining part of the Williams experience. From the moment they arrive, students find themselves in what’s likely the most diverse collection of backgrounds, perspectives and interests they’ve ever encountered. What might differentiate you from the 19 other first-year students in an entry? What perspective(s) would you add to the conversation with your peers? Each question has the same instructions: respond to the prompt in 300 words or fewer. Writing the Williams writing supplement is optional, so you can choose whether you want to answer a question or not. Should I Write an Essay for the Williams Writing Supplement? When you’re working on your Williams College application, you might notice that the Williams Writing Supplement is entirely optional. So should you write an essay? Or skip it altogether? It would be a huge mistake to not write the Williams College supplement. While the instructions do say optional, the statement isn’t really optional. Choosing not to write an essay will make you look like you don’t care that much about being accepted to Williams. Along the same lines, your Williams writing supplement is a great way to show the admissions committee aspects of your personality that aren’t highlighted in the rest of your application. Take that opportunity! Show the admissions committee why you belong on Williams’ campus. What Should I Write About in My Williams College Supplement? Let’s take a look at each of the Williams College supplement questions and discuss what you could write about in each. At Williams we believe that bringing together students and professors in small groups produces extraordinary academic outcomes. Our distinctive Oxford-style tutorial classes- in which two students are guided by a professor in deep exploration of a single topic- are a prime example. Each week the students take turns developing independent work- an essay, a problem set, a piece of art- and critiquing their partner’s work. Focused on close reading, writing and oral defense of ideas, more than 60 tutorials a year are offered across the curriculum, with titles like Aesthetic Outrage, Financial Crises: Causes and Cures, and Genome Sciences: At the Cutting Edge. Imagine yourself in a tutorial at Williams. Of anyone in the world, whom would you choose to be your partner in the class, and why? While it may seem like there are endless ways to answer this question, there are really only two real options: you can pick someone you know personally or you can pick someone you’ve never met, but have always wanted to. Whichever direction you go in, you should make sure to have a specific reason for choosing that person. If you pick someone you know personally, you can use this essay as an opportunity to talk about experiences you’ve had that have greatly affected you. You could, for instance, choose someone you met on a service trip who taught you about hard work or the director of a musical that you participated in that taught you a lot about self confidence. In either of these examples, you’ll be able to talk not only about the influential figure, but about an important part of your life (the service trip or the musical). If you decide to go the celebrity or famous person route, you should make sure to have a real reason why you want to meet that person - a reason that reflects how they influence you. Love isn’t the same as influence - you can love a celebrity but that doesn’t mean they’ve had a huge impact on your life. It’s fine to pick Chrissy Teigen, but only if you talk about how you’d really like her help dissecting a tutorial on social media. If you’re struggling to pick a person, it can be helpful to come up with a tutorial topic that you’d like to participate in first. Having parameters like class topic can be useful for giving you ideas for how to answer the question. Each Sunday night, in a tradition called Storytime, students, faculty and staff gather to hear a fellow community member relate a brief story from their life (and to munch on the storyteller’s favorite homemade cookies). What story would you share? What lessons have you drawn from that story, and how would those lessons inform your time at Williams? While this prompt talks a Williams-specific tradition, Storytime, the question itself is a common one in admissions essays: sharing about a time when you learned an important lesson. To master this prompt, you need to pick a specific experience. It doesn’t need to be earth-shattering or impressive, but it does need to have real significance in your life. You should pick an authentic experience that you actually had - don’t make something up or exaggerate to try to seem more important. Your essay should have a clear narrative arc with a beginning, middle, and end. Make sure to include your takeaways and reflections in the end of the response. Every first-year student at Williams lives in an Entry- a thoughtfully constructed microcosm of the student community that’s a defining part of the Williams experience. From the moment they arrive, students find themselves in what’s likely the most diverse collection of backgrounds, perspectives and interests they’ve ever encountered. What might differentiate you from the 19 other first-year students in an entry? What perspective(s) would you add to the conversation with your peers? This Williams College supplement prompt gives you an opportunity to share more about what makes you unique. Don’t fall into the trap, though, of sharing too much! Pick one specific trait or identity to talk about. You don’t need to talk about every single thing you’ve ever done or liked. In your essay, be sure to talk about how the trait or identity you chose has affected your perspective. Maybe being introverted has let you observe more about other people. Maybe being a member of the LGBTQ+ community has taught you about the importance of respecting others’ differences. Whatever you choose, make sure to fully flesh out how and why that trait has affected your perspective and why that perspective would be valuable to the Williams community. Tips for Writing a Strong Williams College Supplement Essay Writing a strong Williams College supplement essay isn’t just about picking the right prompt to answer. You need to make sure your essay is the best possible example of your work in order to wow the admissions committee. Follow these three tips for writing an amazing Williams supplement essay. #1: Be Authentic The point of a college essay is for the admissions committee to have the chance to get to know you beyond your test scores, grades, and honors. Your admissions essays are your opportunity to make yourself come alive for the essay readers and to present yourself as a fully fleshed out person. You should, then, make sure that the person you’re presenting in your college essays is yourself. Don’t try to emulate what you think the committee wants to hear or try to act like someone you’re not. If you lie or exaggerate, your essay will come across as insincere, which will diminish its effectiveness. Stick to telling real stories about the person you really are, not who you think Williams wants you to be. #2: Play With Form The Williams College supplement essays leave a lot of room open for creative expression - use that! You don’t need to stick to a five paragraph essay structure here. You can play with the length and style of your sentences - you could even dabble in poetry if that makes sense! Whichever form you pick, make sure it fits with the story you’re trying to tell and how you want to express yourself. #3: Proofread and Polish Your Essay Your Williams essay should be the strongest example of your work possible. Before you turn in your application, make sure to edit and proofread your essays. Your work should be free of spelling and grammar errors. Make sure to run your essays through a spelling and grammar check before you submit. It’s a good idea to have someone else read your Williams College supplement essay, too. You can seek a second opinion on your work from a parent, teacher, or friend. Ask them whether your work represents you as a student and person. Have them check and make sure you haven’t missed any small writing errors. Having a second opinion will help your work be the best it possibly can be. Final Thoughts While the Williams College supplement says it’s optional, it’s not really! You should answer the essay as part of your application. When writing your Williams College supplement response, DO: Be authentic and true to yourself. Tell stories that are meaningful to your identity and experience. DON’T: Lie or exaggerate to seem more important. Forget to proofread or polish your essay. What’s Next? Wondering how to ace the Common Application? No problem! We’ve got you covered with tips and tricks to make your application stand out from the crowd. Starting your essay is often the hardest part.If you're unsure where to begin, check out this guide to starting a college essay perfectly, and don't be afraid to just dive right in! If you're applying to Williams College, you're likely applying to other colleges on the East Coast, too. Check out our expert guides to the Duke essay, the Tufts essays, and the Harvard essay. Want to write the perfect college application essay? Get professional help from PrepScholar. Your dedicated PrepScholar Admissions counselor will craft your perfect college essay, from the ground up. We'll learn your background and interests, brainstorm essay topics, and walk you through the essay drafting process, step-by-step. At the end, you'll have a unique essay that you'll proudly submit to your top choice colleges. Don't leave your college application to chance. Find out more about PrepScholar Admissions now: