Question

Solve this simultaneous equation by substitution:

y=x^2-1
y=5-x

Answer 1

We have the following system of equations,

`\left \{ {{y=x^2-1} \atop {y=5-x}} \right.`

We are asked to solve the system of equations using substitution.

What is substitution?

Substitution is a method of solving a system of equations where you solve for one of the equations (pick either equation) for one of the variables (pick one of the variables), then using this new equation to "substitute" back in to the other.

Given this set of equations,
`\left \{ {{y=x^2-1} \atop {y=5-x}} \right.`, notice how both of the equations are already solved for the variable,
`y`. Since both these equations equal
`y`, we can set them equal to each other and solve for,
`x`.

`\left \{ {{y=x^2-1} \atop {y=5-x}} \right.`

`\Longrightarrow 5-x=x^2-1`, substituted "
`5-x`" for
`y` into the top equation.

`\Longrightarrow -x^2-x+6=0`

`-1[\Longrightarrow -x^2-x+6=0]`

`\Longrightarrow x^2+x-6=0`

`\Longrightarrow (x-2)(x+3)=0`

`\Longrightarrow x=2 \ and \ x=-3`

Now take these values of x (
`x=2 \ and \ x=-3`) and plug them into either of the two original equations to solve for
`y`.

When x=2:

`y=5-x;x=2`

`\Longrightarrow y=5-(2)`

`\Longrightarrow y=3`

When x=-3:

`y=5-x;x=-3`

`\Longrightarrow y=5-(-3)`

`\Longrightarrow y=5+3`

`\Longrightarrow y=8`

Thus, we get two solutions to the given system of equations, (-3,8) & (2,3).