Math  /  Algebra

Questiong system of equations graphically on the set y=x2y=3x6\begin{array}{c} y=x-2 \\ y=-3 x-6 \end{array}

Studdy Solution

STEP 1

What is this asking? We need to draw the lines y=x2y = x - 2 and y=3x6y = -3x - 6 on a graph, and find the point where they meet! Watch out! Make sure your lines are straight and accurate, so you can pinpoint the exact intersection point.
A tiny mistake in the drawing can lead to a wrong answer!

STEP 2

1. Draw the first line
2. Draw the second line
3. Find the intersection

STEP 3

The **y-intercept** is where the line crosses the y-axis, which happens when x=0x = 0.
Plugging x=0x = 0 into y=x2y = x - 2, we get y=02=2y = 0 - 2 = \mathbf{-2}.
So our **y-intercept** is at the point (0,2)(0, -2).

STEP 4

The **x-intercept** is where the line crosses the x-axis, which happens when y=0y = 0.
Plugging y=0y = 0 into y=x2y = x - 2, we get 0=x20 = x - 2.
Adding 2 to both sides gives us x=2x = \mathbf{2}.
So our **x-intercept** is at the point (2,0)(2, 0).

STEP 5

Now, **plot** the two points (0,2)(0, -2) and (2,0)(2, 0) on your graph.
Draw a straight line through these points, extending it beyond them in both directions.
Ta-da! The first line is drawn.

STEP 6

Just like before, set x=0x = 0 in y=3x6y = -3x - 6.
This gives us y=306=6y = -3 \cdot 0 - 6 = \mathbf{-6}.
So the **y-intercept** is (0,6)(0, -6).

STEP 7

Set y=0y = 0 in y=3x6y = -3x - 6.
This gives us 0=3x60 = -3x - 6.
Adding 66 to both sides gives us 6=3x6 = -3x.
Dividing both sides by 3-3 gives us x=2x = \mathbf{-2}.
So the **x-intercept** is (2,0)(-2, 0).

STEP 8

**Plot** the points (0,6)(0, -6) and (2,0)(-2, 0) on the same graph as the first line.
Draw a straight line through these points, extending it in both directions.
Awesome, the second line is done!

STEP 9

Look closely at your graph!
Where do the two lines cross?
That's the **point of intersection**, which is the solution to the system of equations.

STEP 10

Carefully determine the x and y values of the intersection point.
It looks like the lines intersect at (1,3)(-1, -3).

STEP 11

The solution to the system of equations is x=1x = \mathbf{-1} and y=3y = \mathbf{-3}.
This means if we plug in x=1x = -1 into both equations, we get y=3y = -3 for both!

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