Linear Equations | Algebra

Introduction

Linear equations form a crucial part of the quantitative reasoning section, primarily emphasizing algebraic understanding and problem-solving skills. These questions sometimes require the manipulation of equations to solve for unknown variables; in other cases, they call on you to interpret the relationships they describe.

Approach Question

What is the slope of the line in the $(x, y)$ -coordinate plane that passes through the point $(- 4, 4)$ and has a $y$ -intercept of $- 8$ ?

A. -2
B. -3
C. -4
D. 4

Explanation

The slope of a line in the coordinate plane is calculated using the formula:

$slope = \frac{Δ y}{Δ x}$

(Note: the $Δ$ symbol means “change in …”)

But at first glance, it might appear that we don’t have enough information to use the slope formula. Aren’t we only given one point, the point $(- 4, 4)$ ? In fact, we do have another point, but in disguise. Anytime a CLT question gives you an intercept, whether it’s an $x$ -intercept or a $y$ -intercept, you have a point where one of the coordinates is zero. The rule is this: for the $x$ -intercept, the $y$ -coordinate is zero, and for the $y$ -intercept, the $x$ -coordinate is zero. This is because on the $x$ -axis, the $y$ value must always be zero, since the points are neither above nor below the axis. And vice versa for the $y$ -axis.

So it turns out we do have a second point: the point $(0, - 8)$ . We can now use the slope formula as follows:

$slope slope slope slope = Δ y /Δ x = (- 8 - 4) / (0 - (- 4) = (- 12) / (4) = - 3$

So the answer is B. Note that in calculating the slope, we chose to begin each “change in” calculation (as in “change in $y$ ”) with the point $(0, - 8)$ . We could have just as easily started with the point $(- 4, 4)$ , which would have looked like this:

$slope slope slope slope = Δ y /Δ x = (4 - (- 8)) / (- 4 - 0) = (12) / (- 4) = - 3$

The result is the same. Be careful to avoid the following two common errors:

Putting “change in $x$ ” in the numerator instead of “change in $y$ ”.
Switching the order of the points as you calculate–for example, starting with the first $y$ -coordinate but the second $x$ -coordinate.

Definitions

Linear Equation: An equation involving only first-degree polynomials. For example, $a x + b = c$ , where $a$ , $b$ , and $c$ are constants (numbers), is a linear equation in one variable.
Variable: A symbol used to represent a number in equations and expressions.
Constant: A value that does not change.
Slope: A measure of the steepness of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on the line.

Topics for Cross-Reference

Systems of linear equations

Variations

Linear equations can also be presented in context, such as in word problems, where translating a scenario into an equation becomes a critical skill.

Strategy Insights

Always isolate the variable to one side of the equation to simplify the solving process.
Remember to perform any operation on both sides of an equation to maintain equality.
Check your solutions by substituting them back into the original equation to ensure they satisfy the equation.
Remember the “reverse rule” for linear equations, which reverses $x$ and $y$ in two primary circumstances: 1) In the case of intercepts, the value that is zero is the opposite of the intercept (so, for the $x$ -intercept, the $y$ value is zero). 2) For reflection across an axis, we negate the opposite variable (for example, if reflecting across the $y$ -axis, we negate the $x$ coordinate while the $y$ stays the same).

Flashcard Fodder

There are several equations and formulas important to memorize if you want to excel on linear equations questions. We have already mentioned the slope formula, which we put in expanded form here:

$slope = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Also, there are the two most prominent forms of the linear equation:

Slope-intercept form: $y = m x + b$
Standard form: $A x + B y = C$

A very handy tool is the formula for slope of a line in standard form, which is simply $\frac{- A}{B}$ . This formula saves you the time of converting standard form into slope-intercept form to find the slope.

Finally, you may have encountered point-slope form during your study of linear equations in school:

$(x - x_{1}) = m (y - y_{1})$

This equation can be helpful when you have only the slope ( $m$ ) and one point ( $x_{1}, y_{1}$ ). If you remember the equation and how to use it, we recommend using it under these circumstances, but if not, you’ll find that you have everything you need between slope-intercept form and standard form.

Finally: regarding parallel and perpendicular lines:

Parallel lines have identical slopes.
Perpendicular lines have negative (or opposite) reciprocal slopes. (Reminder: a reciprocal is equal to $1$ divided by the original number; in practice terms, it’s a fraction flipped upside down.)

Sample Questions

Difficulty 1

If $y = 5 x - 15$ , what is the value of $x$ when $y$ is equal to $10$ ?

A. $x = 15$
B. $x = 10$
C. $x = 5$
D. $x = 0$

(spoiler)

The answer is C. This is a straightforward “plug in” (substitution) question; we just need to be careful for which variable we substitute the $10$ . The question says that it’s the $y$ variable that’s $10$ , so can rewrite the equation as $10 = 5 x - 15$ . We then solve as follows:

$10255 = 5 x - 15 = 5 x = x$

We will frequently remind you, after showing the solution to a problem, that you can always plug in the answers should you choose to do so. However, for a problem with this few algebraic steps, most students will find it faster and more reliable to simply solve the equation.

Difficulty 2

Line $t$ is modeled by the equation $2 x + 4 y = 3$ . Line $s$ is parallel to Line $t$ and has an $x$ -intercept of $4$ … What is the y-intercept of line $s$ ?

A. 0
B. 2
C. 4
D. 6

(spoiler)

The answer is B. The UnCLES method reminds us to note key terms, and the term “parallel” is our important starting point for this question. It tells us that line $s$ has the same slope as line $t$ . We can use the helpful formula $- A / B$ to find the slope of line $t$ ; its slope, and therefore also the slope of line $s$ , is $- \frac{1}{2}$ . The next step involves using the ever-helpful slope-intercept form: $y = m x + b$ . Whenever we have the slope of a line and a point on the line, we can plug in both to slope-intercept form, leaving the only unknown as b. Remember that the $x$ -intercept of $4$ is the point $(4, 0)$ , we have the following after substituting the slope for $m$ and the point’s coordinates for $x$ and $y$ :

$0 = (- \frac{1}{2}) (4) + b$

Simplifying and solving gives us:

$02 = - 2 + b = b$

If you’re wondering about alternate approaches to this question, you might consider graphing if you’re confident graphing lines based on one point and the slope. You could graph this line and discover that it crosses the $y$ -axis at $2$ . The challenge is to graph the accurately enough on your scratch paper to ensure that your answer is correct. If you plan to solve some linear equation questions on test day, we recommend you begin practicing with graph paper now. The CLT permits the use of graph paper for your scratch paper; you may also use lined paper and blank paper.

Difficulty 3

Which of the following lines is perpendicular to a line that passes through $(- 1, 4)$ and $(2, - 5)$ ?

A. $y = - 3 x + 7$
B. $y = - \frac{1}{3} x - 3$
C. $y = \frac{1}{3} x + 4$
D. $y = 9 x - 1$

(spoiler)

The answer is C. Don’t confuse parallel lines with perpendicular lines; parallel lines would have the same slope as the original line (and you can bet they’ll include a trap answer to that effect)! For this question, we’re looking for the negative reciprocal of the original line. So what’s the slope of the original line? Applying the slope formula, we get:

$slope slope slope = (4 - (- 5)) / (- 2 - 1) = (9) / (- 3) = - 3$

The negative reciprocal of $- 3$ is $\frac{1}{3} .$

Note that we don’t have to take further steps because each of the answers is in slope-intercept form and only one answer has the correct slope; we can tell that by looking for the $m$ in $y = m x + b$ …

Alicia skis down a ski run with a slope of $- \frac{1}{4}$ . If the vertical drop of the run is $75$ feet, what is the horizontal length of the ski run, in feet?

A. $300$
B. $150$
C. $75$
D. $18.75$

(spoiler)

The answer is A. A word problem like this calls for the slope formula, but we need to set it up carefully. The vertical drop, given as $75$ , corresponds to rise in the $\frac{r i se}{r u n}$ formula. This means that, in the proportion, $75$ must be in the numerator. In addition, we can avoid later confusion about the sign of the result by noting that $75$ should be negative since there is a vertical drop. So the problem unfolds like this:

$\frac{- 1}{4} = \frac{- 75}{x} - x = - 300 x = 300$

In the steps above, we placed the negative sign in front of the numerator to make clear that the rise or vertical drop is really what’s negative.

Finally, note that answer choice $D$ results from setting up the proportion upside down, with the rise in the numerator and the run in the denominator.

Difficulty 4

A point $(- 2, 6)$ is reflected across the $y$ -axis, then rotated $180°$ . If the resulting point is on a line that also includes the point $(4, - 9)$ , what is the equation of that line?

A. $y = - 2 x - 7$
B. $y = - \frac{1}{2} x - 7$
C. $y = \frac{1}{2} x - 3$
D. $y = \frac{3}{2} x + 1$

(spoiler)

The answer is B. As shown under Flashcard Fodder, a reflection negates the opposite variable from the axis over which it is reflected. Since this point is reflected over the $y$ -axis, we negate the $x$ -coordinate, turning it positive. The new point is $(2, 6)$ . What about the rotation? If you need to visualize this, graph the point $(2, 6)$ and draw a circle going through it with its center at the origin. The new point should travel exactly halfway around that circle. Accordingly, the rotation will negate both coordinates, so that our point is now at $(- 2, - 6)$ . Now we can make the point we just identified interact with the other given point using the slope formula. We’ve laid that out already twice in this lesson; try it on your own and see what you get. You should find the slope connecting the two lines is $- \frac{1}{2}$ . We don’t need to concern ourselves with the $y$ -intercept, because only one answer matches the slope we’ve discovered. (However, checking the $y$ -intercept for answer B to make sure it fits our data would be an excellent way to make sure we haven’t made a mistake along the way. Problems with many steps should always be double-checked if time permits!)

Difficulty 5

Line $j$ is represented by the equation $3 x - 7 y = - 7$ . Line $k$ is represented by the equation $y = - \frac{1}{7} x + 5$ . If Line $l$ passes through the intersection point of Lines $j$ and $k$ and the point $(- 11, 0)$ , what is the slope of Line $l$ ?

A. $- \frac{8}{3}$
B. $- 1$
C. $\frac{2}{9}$
D. $2$

(spoiler)

The answer is C. We first need to find the intersection point of lines $j$ and $k$ . If both had been in slope-intercept form, we would have been able to set the $m x + b$ portions equal to each other, but one of these equations is in standard form. Since $y$ is isolated in the second equation, we can plug in the right side of that equation for $y$ so that we end up with $3 x - 7 (- \frac{1}{7} x + 5) = - 7$ . Simplifying gives us $3 x + 1 x - 35 = - 7$ , then $4 x = 28$ , then $x = 7$ . Plugging $7$ in for $x$ in either equation, we discover that $y$ is $4$ . With this point in mind, we can find the equation for Line $l$ starting with $(7, 4)$ and the new point $(- 11, 0)$ . Using the slope formula, we find $\frac{4 - 0}{7 - ( - 11 )}$ , which comes to $\frac{4}{18}$ , which reduces to $\frac{2}{9}$ .

For Reflection

What strategies do you find most effective when solving linear equations?
Rate the difficulty of linear equations for you from 1 (easy) to 5 (challenging). How comfortable are you with manipulating equations?
Have you applied linear equations to real-world problems? If so, describe how understanding linear equations helped you.