A system of linear equations is shown on the graph.
The graph shows two intersecting lines. One line passes through the points (negative 3, negative 4), (0, negative 2), and (3, 0). The other line passes through the points (negative 3, negative 4), (negative 1, 0), (0, 2), and (1, 4).
What is the solution to the system of equations? Type the coordinates in the boxes below.
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,
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Based on the information given, the solution to the system of equations is the point of intersection between the two lines. To find this point, we can find the equations of the lines using the given points and then find the point of intersection.
First, let's find the equation of the line that passes through the points (-3, -4), (0, -2), and (3, 0).
We can find the slope (m) of this line using the formula:
m = (y2 - y1) / (x2 - x1), where (x1, y1) = (-3, -4) and (x2, y2) = (3, 0).
m = (0 - (-4)) / (3 - (-3)) = 4 / 6 = 2/3
Using the point-slope form of a line, y - y1 = m(x - x1), we can choose any point on the line to plug in for (x1, y1) and solve for the equation.
Let's use the point (0, -2):
y - (-2) = (2/3)(x - 0)
y + 2 = (2/3)x
y = (2/3)x - 2
Next, let's find the equation of the line that passes through the points (-3, -4), (-1, 0), (0, 2), and (1, 4).
Using the same method, we can find the slope (m) of this line:
m = (4 - (-4)) / (1 - (-3)) = 8 / 4 = 2
Using the point-slope form, we'll use the point (-1, 0) as (x1, y1):
y - 0 = 2(x - (-1))
y = 2x + 2
Now we have the equations of the two lines:
Line 1: y = (2/3)x - 2
Line 2: y = 2x + 2
To find the point of intersection, we can set the two equations equal to each other and solve for x:
(2/3)x - 2 = 2x + 2
2/3x - 2x = 2 + 2
(2 - 6/3)x = 4
(-4/3)x = 4
Multiplying both sides by -3/4 to isolate x:
x = -3
Now we can plug this value of x back into either of the equations to find y.
Using the equation y = (2/3)x - 2:
y = (2/3)(-3) - 2
y = -2 - 2
y = -4
Therefore, the solution to the system of equations is the point (-3, -4).
(-3, -4)