Question

A system of linear equations includes the line that is created by the equation y = x+ 3, graphed below, and the line through the points (3, 1) and (4, 3).

On a coordinate plane, a line goes through (0, 3) and (2, 5).
What is the solution to the system of equations?
(–1, 2)
(1, 3)
(8, 11)
(9, 12)

Answer 1

To solve the system of linear equations that includes the line given by \( y = x + 3 \) and the line through the points \( (3, 1) \) and \( (4, 3) \), we first need to determine the equation of the second line.

**Step 1: Find the equation through points (3, 1) and (4, 3).**

1. Calculate the slope of the line:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{4 - 3} = \frac{2}{1} = 2
\]

2. Use the point-slope form of the equation of a line (\( y - y_1 = m(x - x_1) \)):
\[
y - 1 = 2(x - 3)
\]
Simplifying this:
\[
y - 1 = 2x - 6 \\
y = 2x - 5
\]

Now we have the two equations:
1. \( y = x + 3 \)
2. \( y = 2x - 5 \)

**Step 2: Set the equations equal to find the intersection.**
\[
x + 3 = 2x - 5
\]

**Step 3: Solve for \( x \).**
\[
3 + 5 = 2x - x \\
8 = x
\]

**Step 4: Substitute \( x = 8 \) back into either equation to find \( y \).** We can use \( y = x + 3 \):
\[
y = 8 + 3 = 11
\]

Thus, the solution to the system of equations is \( (8, 11) \).

**Final Step: Verify the answer.**
We have our potential solutions as:
- (–1, 2)
- (1, 3)
- (8, 11)
- (9, 12)

Among these, the intersection we found, which is \( (8, 11) \), matches with one of the provided options.

Thus, the solution to the system of equations is \( (8, 11) \).