To describe the equations \( y = x + 3 \) and \( -4x + 4y = 28 \) as a graph, we first need to rewrite both equations in a standard format and then analyze them.
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Equation 1: \( y = x + 3 \)
This is already in slope-intercept form \( y = mx + b \) where \( m \) (slope) is 1 and \( b \) (y-intercept) is 3.
- The slope indicates that for every unit increase in \( x \), \( y \) increases by 1.
- The y-intercept indicates that the line crosses the y-axis at \( (0, 3) \).
- To plot this, you can choose other points like:
- When \( x = 0, y = 3 \) (point \( (0,3) \))
- When \( x = 2, y = 5 \) (point \( (2,5) \))
- When \( x = -3, y = 0 \) (point \( (-3,0) \))
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Equation 2: \( -4x + 4y = 28 \)
First, let's convert this to slope-intercept form. We can do this by solving for \( y \): \[ 4y = 4x + 28 \] \[ y = x + 7 \] This equation is also in the form \( y = mx + b \) where \( m = 1 \) and \( b = 7 \).
- Similar to the first equation, this line also has a slope of 1.
- The y-intercept is 7, meaning the line crosses the y-axis at \( (0, 7) \).
- You can find additional points to plot:
- When \( x = 0, y = 7 \) (point \( (0,7) \))
- When \( x = 2, y = 9 \) (point \( (2,9) \))
- When \( x = -3, y = 4 \) (point \( (-3,4) \))
Graphing the Lines:
- Both lines have the same slope (1), which means they are parallel lines.
- The first line (from \( y = x + 3 \)) has a lower y-intercept and thus is always below the second line (from \( y = x + 7 \)) at any given point \( x \).
Summary of the Graph:
- Both lines are straight and have a slope of 1, indicating they rise at the same angle from left to right.
- The first line will intersect the y-axis at (0, 3) and the second line at (0, 7).
- The lines do not intersect since they are parallel and maintain a constant distance apart.
This information can be used to sketch a graph showing these two parallel lines, providing a visual representation of their relationship.
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