AI-generated answer
To solve the system of equations:
y = -5x + 3
y = 2x^2 - x - 3
We can use the method of substitution or elimination. Let's use the substitution method in this case.
Step 1: Solve one equation for one variable in terms of the other variable.
From the first equation, we can solve for y:
y = -5x + 3
Step 2: Substitute the expression for y into the other equation.
Replace y in the second equation with the expression -5x + 3:
-5x + 3 = 2x^2 - x - 3
Step 3: Rearrange the equation into standard form.
Move all terms to one side to get a quadratic equation:
2x^2 - x + 5x - 6 = 0
2x^2 + 4x - 6 = 0
Step 4: Solve the quadratic equation.
We can either factor, complete the square, or use the quadratic formula to solve the equation. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 4, and c = -6.
x = (-4 ± √(4^2 - 4(2)(-6))) / (2(2))
x = (-4 ± √(16 + 48)) / 4
x = (-4 ± √64) / 4
x = (-4 ± 8) / 4
This gives us two possible solutions for x:
x1 = (-4 + 8) / 4 = 1
x2 = (-4 - 8) / 4 = -3
Step 5: Substitute the values of x into one of the original equations to find the corresponding y values.
Using the first equation, let's substitute x = 1:
y = -5(1) + 3
y = -5 + 3
y = -2
Using the first equation again, let's substitute x = -3:
y = -5(-3) + 3
y = 15 + 3
y = 18
So, the solutions to the system of equations are:
(x, y) = (1, -2) and (-3, 18)
These are the points where the two equations intersect on the coordinate plane.