What does it mean to solve a system of equations like y=x^2+4x-5 and y=x-1 graphically?
graph the first equation, which is parabola, and has x-intercepts of -5 and 1, opening upwards
the second equation is a straight line with slope 1 and y-intercept of -1
my rough sketch shows them intersecting at two points.
the intersection of the the two graphs is the "solution to the system" of equations.
thanks
To solve a system of equations graphically, you need to find the points where the graphs of the equations intersect. In this case, the system of equations is y = x^2 + 4x - 5 and y = x - 1.
To graphically solve this system of equations, you can follow these steps:
1. Plot the first equation, y = x^2 + 4x - 5, on a coordinate plane. To do this, you can create a table of x and y values and plot the corresponding points. Choose a range of x values and calculate the corresponding y values using the equation. For example:
- For x = -3, y = (-3)^2 + 4(-3) - 5 = 9 - 12 - 5 = -8
- For x = -2, y = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9
- For x = -1, y = (-1)^2 + 4(-1) - 5 = 1 - 4 - 5 = -8
- For x = 0, y = (0)^2 + 4(0) - 5 = 0 - 0 - 5 = -5
- For x = 1, y = (1)^2 + 4(1) - 5 = 1 + 4 - 5 = 0
- For x = 2, y = (2)^2 + 4(2) - 5 = 4 + 8 - 5 = 7
- For x = 3, y = (3)^2 + 4(3) - 5 = 9 + 12 - 5 = 16
Plot these points and connect them to form a curve.
2. Plot the second equation, y = x - 1, on the same coordinate plane. Again, create a table of x and y values and plot the corresponding points:
- For x = -3, y = -3 - 1 = -4
- For x = -2, y = -2 - 1 = -3
- For x = -1, y = -1 - 1 = -2
- For x = 0, y = 0 - 1 = -1
- For x = 1, y = 1 - 1 = 0
- For x = 2, y = 2 - 1 = 1
- For x = 3, y = 3 - 1 = 2
Plot these points and connect them to form a straight line.
3. Look for the intersection point(s) of the two graphs. These points represent the solutions to the system of equations. In this case, the two graphs intersect at the point (1, 0).
Therefore, the solution to the system of equations y = x^2 + 4x - 5 and y = x - 1 graphically is x = 1 and y = 0.