Question

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Solve the following system of equations and show all work. y = −x2 + 4 y = 2x + 1

Answer 1

The solutions to the given system of equations are:

• x = 1, y = 3
• x = -3, y = -5

Explanation:

Given system of equations:

`\begin{cases}y=-x^2+4\\y=2x+1\end{cases}`

To solve the given system of equations, substitute the second equation into the first equation:

`\implies 2x+1=-x^2+4`

Add x² to both sides:

`\implies x^2+2x+1=4`

Subtract 4 from both sides:

`\implies x^2+2x-3=0`

Factor the quadratic equation:

`\implies x^2+3x-x-3=0`

`\implies x(x+3)-1(x+3)=0`

`\implies (x-1)(x+3)=0`

Apply the zero product property:

`(x-1)=0 \implies x=1`

`(x+3)=0 \implies x=-3`

Substitute the found values of x into the second equation and solve for y:

`x=1 \implies y=2(1)+1=3`

`x=-3 \implies y=2(-3)+1=-5`

Therefore, the solutions to the given system of equations are:

• x = 1, y = 3
• x = -3, y = -5