To determine the number of solutions for the system of equations and to describe their graph, let's analyze the equations you've provided:
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The first equation is \( x - 2y = 8 \).
- To convert this into slope-intercept form (y = mx + b), we can solve for \( y \): \[ x - 2y = 8 \ -2y = -x + 8 \ y = \frac{1}{2}x - 4 \]
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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 \( -4 \).