Question

Solve the following system of equations graphically on the set of axes below. y, equals, x, minus, 4 y=x−4 x, plus, 2, y, equals, 10 x+2y=10

Answer 1

To solve the system of equations graphically, we first need to rewrite the given equations in slope-intercept form (y = mx + b) to identify their slopes and intercepts.

1. **First equation:** \( y = x - 4 \)

This is already in slope-intercept form, where the slope \( m = 1 \) and the y-intercept \( b = -4 \).

2. **Second equation:** \( x + 2y = 10 \)

To convert this to slope-intercept form, solve for \( y \):

\[
2y = -x + 10
\]
\[
y = -\frac{1}{2}x + 5
\]

This gives us a slope of \( m = -\frac{1}{2} \) and a y-intercept of \( b = 5 \).

Now we can plot both equations on a graph:

**For the first equation, \( y = x - 4 \)**:
- When \( x = 0 \): \( y = 0 - 4 = -4 \) (point: (0, -4))
- When \( x = 4 \): \( y = 4 - 4 = 0 \) (point: (4, 0))
- When \( x = 6 \): \( y = 6 - 4 = 2 \) (point: (6, 2))

**For the second equation, \( y = -\frac{1}{2}x + 5 \)**:
- When \( x = 0 \): \( y = -\frac{1}{2}(0) + 5 = 5 \) (point: (0, 5))
- When \( x = 10 \): \( y = -\frac{1}{2}(10) + 5 = 0 \) (point: (10, 0))
- When \( x = 2 \): \( y = -\frac{1}{2}(2) + 5 = 4 \) (point: (2, 4))

Now, you can plot these points on a graph.

After plotting, you can draw the lines for both equations. The point where the two lines intersect is the solution to the system of equations.

**Finding the intersection algebraically:**
We can also solve it algebraically to confirm the graphical solution:

1. From the first equation: \( y = x - 4 \)
2. Substitute \( y \) in the second equation:
\[
x + 2(x - 4) = 10
\]
\[
x + 2x - 8 = 10
\]
\[
3x - 8 = 10
\]
\[
3x = 18
\]
\[
x = 6
\]

Now substitute \( x = 6 \) back into the first equation to find \( y \):
\[
y = 6 - 4 = 2
\]

Thus, the intersection point, which is the solution to the system of equations, is \((6, 2)\).

**Final Solution:** The solution to the system of equations is \( (6, 2) \).