## High School Algebra I Unlocked (2016)

### Chapter 4. Equations and Inequalities

### Lesson 4.4. Two-Variable Linear Equations

Up until now, we’ve only discussed equations with a single variable; for example, *x* = 2 or *y* + 3 = 2*y*. However, in algebra, you will encounter tons of questions that require you to manipulate and graph two-variable equations.

**Graphing and the Coordinate Plane**

**Graphing** is a way of representing a point in two dimensions on what is known as a **Cartesian grid** or **coordinate plane**. Here’s the Cartesian grid:

**Here is how you may see equivalent expressions on the ACT.**

−*x*^{3} + 24*x* + 75 + 34*x*^{3} − 36*x* is equivalent to:

F. 99*x*

G. −35*x*^{3}

H. 24*x* + 75

J. 33*x*^{3} − 12*x* + 75

K. 35*x*^{3} + 12*x* + 75

Graphing is a way of assigning points to this grid. Every point has two numbers assigned to it: an *x*-coordinate and a *y*-coordinate. Let’s take point *A* (3, 1) in the following graph. The first number is considered the *x*-coordinate, and the second number is the *y*-coordinate. To plot this point on the graph, we start at (0, 0), also referred to as the **origin**, and count over three to the right on the *x*-axis, and then count up one place parallel to the *y*-axis. To find point *B* (5, 4), we move five places to the right of the origin on the *x*-axis, and then up four places parallel to the *y*-axis.

If the *x*-coordinate is negative, move to the left along the *x*-axis. If the *y*-coordinate is negative, then you move downward parallel to the *y*-axis. Look at the graph above to see how points (−5, 4) and (−3, −2) are plotted.

**Slope-Intercept Form**

In order to graph linear equations with ease, you need to be familiar with the **slope-intercept form** of a line:

*y* = *mx* + *b*

When an equation is in slope-intercept form, *m* represents the **slope**, or the measure of the steepness of a line, and *b* represents the ** y-intercept**, or the

*y*-coordinate of the point on the line that crosses the

*y*-axis and has an

*x*-coordinate of 0. To find the

**, set**

*x*-intercept*y*equal to zero and solve for

*x*.

The slope-intercept form of a line is *y* = *mx* + *b*.

*m* = Slope *b* = *y*-intercept

The *x*-intercept is found by setting *y* = 0 and solving for *x*.

As shown in the chart below, the slope of a line can be increasing, decreasing, zero, or undefined, depending on the value of *m*.

**Graphs of Single-Variable Linear Equations**

If you’re wondering why we didn’t cover graphs of single-variable linear equations, consider the equation *x* = 4. This equation indicates that *x* = 4 for *all* values of *y* and, therefore, is a straight vertical line. For all equations that are in the form of *x* = *any number*, the slope is undefined.

Conversely, the equation *y* = 2 indicates that *y* = 2 for *all* values of *x* and, in turn, is a straight horizontal line. For all equations that are in the form of *y* = *any number*, the slope is 0. On the next page, you can see the graphs of both *x* = 4 and *y* = 2.

Start by grouping similar terms. Here, you have −*x*^{3} + 34*x*^{3} = 33*x*^{3}. You can eliminate (F), (G), and (H) because they do not have the term 33*x*^{3} in their expressions. Next, you have 24*x* − 36*x* = −12*x*. The only answer choice that has the term −12*x* is (J).

Alternatively, you could plug in a value for *x*, find the value of −*x*^{3} + 24*x* + 75 + 34*x*^{3} −36*x*. Since there are exponents, pick a small number, like 2. When you set *x* = 2, the equation −*x*^{3} + 24*x* + 75 + 34*x*^{3} − 36*x* becomes −2^{3} + 24(2) + 75 + 34(2)^{3} − 36(2). This simplifies to −8 + 48 + 75 + 272 − 72 = 315. Now, plug *x* = 2 into the answer choices, looking for the value of 315. Only (J), 33(2)^{3} − 12(2) + 75 = 264 − 24 + 75 = 315, provides that value.

Consider the equation *y* = 2*x* + 3. Start out by identifying the *y*-intercept, which is 3. Since the *y*-intercept is 3, the graph includes the point (0, 3). This can be plotted on the graph, as shown.

The slope is 2, and since slope is equal to *rise/run*, the *y*-coordinate increases by 2 as the *x*-coordinate increases by 1. Use this to plot additional points. Start with point (0, 3) and increase the *x*-coordinate by 1 and *y*-coordinate by 2 to get (1, 5). Do this again to get (2, 7) and (3, 9). Similarly, you can go backwards. Go back to the point (0, 3) and decrease the *x*-coordinate by 1 and the *y*-coordinate by 2 to get (−1, 1), and again to get (−2, −1) and (−3, −3). Plot these points as well.

Finally, draw a line that goes through all these points, as shown in the following graph.

This line is a graphical representation of the two-variable linear equation *y* = 2*x* + 3.

Slope can also be found from two points on a coordinate plane using the following formula:

Imagine that you were given the following graph, but *not* given the equation for the line. How would you find the slope?

If you know two points on a coordinate plane, the slope of a line can also be found by using the formula . In the previous graph, there are points at (0, 3) and (−2, −1). You can find the slope, *m*, by substituting those points into the following formula:

Thus, the line in this graph has a slope of 2. Note that the slope of 2 aligns with the value of *m* in the equation *y* = 2*x* + 3.

Now that we’ve learned about two-variable equations, graphs, and slope, let’s try a few questions that deal with these concepts.

**EXAMPLE **

**Rewrite the equation of the line 4 x + 12 = 2y in slope-intercept form, and identify the x- and y-intercepts of the equation.**

This question asks you to rewrite the equation in slope-intercept form, or *y* = *mx* + *b*. Start by rewriting the equation with the *y*-term to the left of the equals sign and the other terms to the right of the equals sign.

4*x* + 12 = 2*y*

2*y* = 4*x* + 12

Next, divide both sides of the equation by 2.

2*y* = 4*x* + 12

*y* = 2*x* + 6

That’s it! The equation 4*x* + 12 = 2*y* in slope-intercept form is *y* = 2*x* + 6. Now you need to identify the *x*- and *y*-intercepts of the equation. In order to find the *x*-intercept, set *y* = 0 and solve for *x*:

*y* = 2*x* + 6

0 = 2*x* + 6

2*x* = −6

*x* = −3

Excellent! You found that the *x*-intercept is (−3, 0). Now you need to find the *y*-intercept, which is the value of *b* when an equation is written in slope-intercept form, *y* = *mx* + *b*. Since the equation 4*x* + 12 = 2*y* has already been converted to slope-intercept form as *y* = 2*x* + 6, the *y*-intercept occurs at (0, 6).

Therefore, the equation 4*x* + 12 = 2*y* is written in slope-intercept form as *y* = 2*x* + 6, the *x*-intercept of the equation is (−3, 0), and the *y*-intercept is (0, 6).

Let’s build on this question.

**EXAMPLE **

**What is the slope of line 4 x + 12 = 2y ?**

You probably noticed that the equation here is the same one from __Example 11__. You should have also noticed that the slope is not easily identifiable in the equation 4*x* + 12 = 2*y*. So, as before, you will need to start by rewriting the equation in slope-intercept form.

4*x* + 12 = 2*y*

2*y* = 4*x* + 12

Next, divide both sides of the equation by 2.

2*y* = 4*x* + 12

*y* = 2*x* + 6

Now that the equation is in slope-intercept form, you can find the slope. Recall that when an equation is in *y* = *mx* + *b* form, the slope is equal to *m*. Therefore, the slope of the line is 2.

Now, you may be wondering if there’s another way to find the slope of a line. You’re in luck! If you have no desire to rewrite the equation in slope-intercept form, you can determine the slope of a line when it is written as a linear equation, or in the form *ax* + *by*+ *c* = 0. In this scenario, the slope is equal to −.

The slope of a line in the form *ax* + *by* + *c* = 0 is equal to −

Let’s look at the previous question again.

**EXAMPLE **

**What is the slope of the line 4 x + 12 = 2y ?**

Instead of rewriting the equation in slope-intercept form, let’s rewrite it in the linear form of an equation, or *ax* + *by* + *c* = 0. Start by subtracting 2*y* from both sides of the equation.

4*x* + 12 = 2*y*

4*x* − 2*y* + 12 = 0

Next, find the slope of the line by calculating −.

*a* = 4 and *b* = −

No matter how you crack it, the slope of the line is 2.

Now let’s walk through a question that tests your ability to identify visual representations of a linear equation.

**EXAMPLE **

**Identify the graph that represents the line 2 x + 3y = 6. Provide the slope, x-intercept, and y-intercept of the equation.**

**A)**

**B)**

**C)**

**D)**

Where to begin? Start by putting the equation into *y* = *mx* + *b* form.

2*x* + 3*y* = 6

3*y* = −2*x* + 6

*y* = −*x* + 2

Now that the equation is in slope-intercept form, you can determine the slope and intercepts of the equation. Based on the equation, the slope of *y* = −*x* + 2 is −, and the *y*-intercept of the equation is (0, 2). Next, find the *x*-intercept by setting *y* = 0 and solving for *x*:

2*x* + 3*y* = 6

2*x* + 3(0) = 6

2*x* = 6

*x* = 3

Therefore, this equation has an *x*-intercept of (3, 0). Now that you have the slope and intercepts, you can identify the graph represented by the equation 2*x* + 3*y* = 6. Both (A) and (D) depict positive slopes and can thus be eliminated. Since we know that this equation has a *y*-intercept of 2, (B) can be eliminated. The only graph that represents an equation with a slope of −2/3, a *y*-intercept of (0, 2), and an *x*-intercept of (3, 0) is (C).

**Here is how you may see slope on the ACT.**

What is the slope of the line based on the equation 5*x* − *y* = 7*x* + 6 ?

F. −2

G. 0

H. 2

J. 6

K. −6