To solve the system of equations y = x + 2 and y = 2x + 7, we can use graphing or create a table of values for both equations.
First, let’s find some points for each equation.
For the first equation: y = x + 2
- If x = 0, then y = 0 + 2 = 2. Point: (0, 2)
- If x = 1, then y = 1 + 2 = 3. Point: (1, 3)
- If x = -1, then y = -1 + 2 = 1. Point: (-1, 1)
For the second equation: y = 2x + 7
- If x = 0, then y = 2(0) + 7 = 7. Point: (0, 7)
- If x = 1, then y = 2(1) + 7 = 9. Point: (1, 9)
- If x = -1, then y = 2(-1) + 7 = 5. Point: (-1, 5)
Now, we can graph both sets of points:
- For the line of y = x + 2, plot the points (0,2), (1,3), and (-1,1). This line has a slope of 1.
- For the line of y = 2x + 7, plot the points (0,7), (1,9), and (-1,5). This line has a steeper slope of 2.
Once we plot these lines on the same graph, we can visually find the intersection point of the two lines, which represents the solution to the system of equations.
Alternatively, we could also set the two equations equal to each other to find the intersection algebraically:
x + 2 = 2x + 7
Solving for x:
- x + 2 – 2x = 7
- -x + 2 = 7
- -x = 7 – 2
- -x = 5
- x = -5
Now substitute x back into either equation to find y:
y = -5 + 2 = -3
Thus, the solution to the system of equations is (-5, -3).