To solve the system of equations by graphing, we have the following two equations:
- \( y = x + 3 \)
- \( -4x + 4y = 28 \)
Step 1: Graph the First Equation
The first equation, \( y = x + 3 \), is in slope-intercept form (y = mx + b), where the slope (m) is 1 and the y-intercept (b) is 3.
- Start at the point (0, 3) on the y-axis.
- From there, rise 1 unit up and run 1 unit to the right to get to (1, 4).
- Connect these points to form a straight line.
Step 2: Graph the Second Equation
To graph the second equation, we can rearrange it into slope-intercept form. Start with:
\[ -4x + 4y = 28 \]
Divide everything by 4:
\[ -y + x = 7 \]
Or rearranged:
\[ y = x - 7 \]
Now, this line has a slope of 1 and a y-intercept of -7.
- Start at (0, -7) on the y-axis.
- From there, rise 1 unit up and run 1 unit to the right to get to (1, -6).
- Connect these points to make a straight line.
Step 3: Analyze the Graph
When you graph both equations on the same piece of paper:
- The line from the first equation ( \( y = x + 3 \) ) will cross the y-axis at the point (0, 3) and will slope upwards.
- The line from the second equation ( \( y = x - 7 \) ) will cross the y-axis at (0, -7) and will also slope upwards but starts much lower.
Step 4: Find the Intersection
When you look at the lines, they will intersect at one point. This point is the solution to the system of equations.
In summary, the graph of both equations shows two straight lines that cross each other. This means there is one solution to the system of equations, which is the point where they intersect. You can find this point by looking at the graph carefully or by solving the equations mathematically to find the exact point.