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The line graphed on the grid represents the first of two equations in a system of linear equations.
 


If the graph of the second equation in the system passes through the points (−12,20)
 and (4,12)
, which statement is true?
A.The only solution to the system is (0,14)
.
B.The system has an infinite number of solutions.
C.The system has no solution.
D.The only solution to the system is (10,5)
.

Let's solve this by analyzing the properties of the lines described by both equations in the system.

### Step 1: Determine the Equation of the Given Line
The line in the graph passes through two clearly identifiable points:
1. Point \( (20, 0) \)
2. Point \( (0, 20) \)

To find the equation of this line, we can determine the slope \( m \) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting \( (x_1, y_1) = (20, 0) \) and \( (x_2, y_2) = (0, 20) \):
\[
m = \frac{20 - 0}{0 - 20} = \frac{20}{-20} = -1
\]

Now, using the point-slope form \( y = mx + b \) and the point \( (0, 20) \), where \( b = 20 \):
\[
y = -x + 20
\]

So, the equation of the line already graphed is:
\[
y = -x + 20
\]

### Step 2: Determine the Equation of the Second Line
The second line in the system passes through the points \( (-12, 20) \) and \( (4, 12) \). We can find its slope \( m \) similarly:
\[
m = \frac{12 - 20}{4 - (-12)} = \frac{-8}{16} = -\frac{1}{2}
\]

Using the point-slope form \( y = mx + b \) and the point \( (4, 12) \), we can solve for \( b \):
\[
12 = -\frac{1}{2}(4) + b
\]
\[
12 = -2 + b
\]
\[
b = 14
\]

So, the equation of the second line is:
\[
y = -\frac{1}{2}x + 14
\]

### Step 3: Determine if There’s an Intersection (Solution to the System)
To find the solution to the system, set the two equations equal to each other and solve for \( x \):
\[
-x + 20 = -\frac{1}{2}x + 14
\]

Add \( \frac{1}{2}x \) to both sides:
\[
-\frac{1}{2}x + 20 = 14
\]

Subtract 20 from both sides:
\[
-\frac{1}{2

The line graphed on the grid represents the first of two equations in a system of linear equations. If the graph of the second equation in the system passes through the points (−12,20) and (4,12), which statement is true?
A. The only solution to the system is (0,14).
B. The system has an infinite number of solutions.
C. The system has no solution.
D. The only solution to the system is (10,5).

Learn how to solve a system of linear equations using graph analysis and slope calculation. This problem involves finding equations of lines and determining if they intersect. Suitable for Grades 8-10 students.

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