QuestionGraph the linear equation.
17. $2 x-3 y=9$
18. $-2 y-4=4$

Studdy Solution

STEP 1

What is this asking? We're being asked to draw the lines described by two equations on a graph! Watch out! Don't forget, linear equations make straight lines on graphs, and we only need two points to define a line!

STEP 2

1. Graph the First Equation
2. Graph the Second Equation

STEP 3

Alright, let's rewrite our first equation, $2x - 3y = 9$ , in slope-intercept form, which is $y = mx + b$ .
This form is super helpful because m tells us the slope of the line (how steep it is) and b tells us the y-intercept (where the line crosses the y-axis).

STEP 4

To get there, we want to isolate y.
First, let's subtract $2x$ from both sides of the equation: $2x - 3y - 2x = 9 - 2x$ .
This simplifies to $-3y = -2x + 9$ .

STEP 5

Now, let's divide both sides by $-3$ to get y all by itself: $\frac{-3y}{-3} = \frac{-2x + 9}{-3}$ .
This gives us $y = \frac{2}{3}x - 3$ .

STEP 6

Boom! Now we can see that our slope, m, is $\frac{2}{3}$ and our y-intercept, b, is $-3$ .

STEP 7

To graph this, first plot the y-intercept at the point $(0, -3)$ .
From there, use the slope, which is $\frac{\textbf{rise}}{\textbf{run}}$ .
Our slope is $\frac{2}{3}$ , so from our y-intercept, we rise 2 units up and run 3 units to the right, giving us a second point at $(3, -1)$ .

STEP 8

Draw a straight line through these two points, and there you have it, the graph of $2x - 3y = 9$ !

STEP 9

Time for the second equation: $-2y - 4 = 4$ .
Let's isolate y again!

STEP 10

First, add 4 to both sides: $-2y - 4 + 4 = 4 + 4$ , which simplifies to $-2y = 8$ .

STEP 11

Next, divide both sides by $-2$ : $\frac{-2y}{-2} = \frac{8}{-2}$ .
This gives us $y = -4$ .

STEP 12

Notice, this equation is in the form $y = b$ , where $b = -4$ .
This means it's a horizontal line that crosses the y-axis at $-4$ .

STEP 13

To graph this, simply draw a horizontal line straight through the point $(0, -4)$ .
Every point on this line will have a y-coordinate of $-4$ .

STEP 14

We've successfully graphed both equations!
The first equation, $2x - 3y = 9$ , is a line that passes through the points $(0, -3)$ and $(3, -1)$ .
The second equation, $-2y - 4 = 4$ , is a horizontal line that passes through the point $(0, -4)$ .

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