(a)

Question

Graph and then find the solution for
y=4/5x -2 and y= -x+7

Answer 1

To find the solution to the system of equations given by

1. \( y = \frac{4}{5}x - 2 \)
2. \( y = -x + 7 \)

we can follow these steps:

Step 1: Graph the equations

1. Graph \( y = \frac{4}{5}x - 2 \):

   • This is a linear equation in slope-intercept form (\( y = mx + b \)) where the slope \( m = \frac{4}{5} \) and the y-intercept \( b = -2 \).
   • Start at the point \( (0, -2) \) on the y-axis.
   • From there, use the slope to find another point:
     • Rise \( 4 \) (up) and run \( 5 \) (right) to get to \( (5, 2) \).
   • Plot these points and draw a line through them.

2. Graph \( y = -x + 7 \):

   • This is also a linear equation where the slope \( m = -1 \) and the y-intercept \( b = 7 \).
   • Start at the point \( (0, 7) \) on the y-axis.
   • From there, use the slope to find another point:
     • Rise \( -1 \) (down) and run \( 1 \) (right) to get to \( (1, 6) \).
   • Plot these points and draw a line through them.

Step 2: Find the intersection point

The solution to the system is where the two lines intersect. This can be found by setting the equations equal to each other:

\[ \frac{4}{5}x - 2 = -x + 7 \]

Step 3: Solve for \( x \)

1. Rearranging the equation:

\[ \frac{4}{5}x + x = 7 + 2 \]

2. Converting \( x \) to a fraction with a common denominator (which is \( 5 \)):

\[ \frac{4}{5}x + \frac{5}{5}x = 9 \] \[ \frac{9}{5}x = 9 \]

3. Now, multiply both sides by \( \frac{5}{9} \):

\[ x = 9 \cdot \frac{5}{9} = 5 \]

Step 4: Find \( y \)

Using \( x = 5 \) in either original equation (let's use the first):

\[ y = \frac{4}{5}(5) - 2 = 4 - 2 = 2 \]

Conclusion

The solution to the system of equations is \( (5, 2) \). This means that the lines intersect at the point \( (5, 2) \).

Summary:

• Intersection point: \( (5, 2) \)