AI Tutor
If you shoot a laser pointer across a room toward a curved mirror, where does the beam hit?
Mathematically, that laser beam is a linear equation (a straight line) and the mirror is a quadratic equation (a parabola or circle).
A linear-quadratic system is a pair of equations where one is degree 1 (linear) and the other is degree 2 (quadratic).
How do we find those exact intersection coordinates algebraically without guessing on a graph?

The Substitution Strategy

Algebraic substitution means isolating one variable in the simpler linear equation and plugging that expression directly into the quadratic equation.
Let's solve this system together: y=2xβˆ’1y = 2x - 1 and y=x2βˆ’4y = x^2 - 4.
Because both equations are already equal to yy, set them equal to each other: x2βˆ’4=2xβˆ’1x^2 - 4 = 2x - 1.
Rearrange all terms to one side to form a standard quadratic equation equal to zero: x2βˆ’2xβˆ’3=0x^2 - 2x - 3 = 0.
Factoring gives (xβˆ’3)(x+1)=0(x - 3)(x + 1) = 0, which yields two xx-values: x=3x = 3 and x=βˆ’1x = -1.
Always plug each xx-value back into the linear equation to get its partner yy-value: y=2(3)βˆ’1=5y = 2(3) - 1 = 5 and y=2(βˆ’1)βˆ’1=βˆ’3y = 2(-1) - 1 = -3.