Consider the system of equations, x – 2y = 8 and –2x + 4y = –16.

What is x – 2y = 8 in slope-intercept form?

y = (1/2)x – 4
What is –2x + 4y = –16 in slope-intercept form?

y = (1/2)x – 4
How many solutions will there be?

What will the graph of the system look like?

1 answer

To determine the number of solutions for the system of equations and to describe their graph, let's analyze the equations you've provided:

  1. The first equation is \( x - 2y = 8 \).

    • To convert this into slope-intercept form (y = mx + b), we can solve for \( y \): \[ x - 2y = 8 \ -2y = -x + 8 \ y = \frac{1}{2}x - 4 \]
  2. The second equation is \( -2x + 4y = -16 \).

    • We can similarly convert it to slope-intercept form: \[ -2x + 4y = -16 \ 4y = 2x - 16 \ y = \frac{1}{2}x - 4 \]

Both equations simplify to the same slope-intercept form: \[ y = \frac{1}{2}x - 4 \]

Determining the Number of Solutions

Since both equations represent the same line, the system has infinitely many solutions. Each equation describes the same relationship between \( x \) and \( y \). This means they are dependent equations.

Graph of the System

The graph of the system will consist of a single line described by either of the equations: \( y = \frac{1}{2}x - 4 \). Since both equations are identical, they will overlap perfectly on the graph.

Summary

  • The equations of the lines are identical.
  • There are infinitely many solutions to the system of equations.
  • The graph will show a single line with a slope of \( \frac{1}{2} \) and a y-intercept of \( -4 \).