Solution by Substitution

Khan Academy: Solving by Substitution

Making y=y (Skill #3)

When two lines are given in slope intercept form, it is a simple enough procedure to substitute the second expression for y into the first equation.

Example:

Suppose we are looking for the one ordered pair that satisfies both equations:

    \[\begin{cases}y=-x+16\\ y=3x-8 \end{cases}\]

 

We are looking for an ordered pair (x, y) that satisfies both equations – that is, when we find the solution, the y coordinate is the same for both equations. The condition that we use to solve this system is that for some value x,

    \[-x+16=3x-8\]

That is, instead of writing y on the left, we substituted the other expression for y. Now we have a reasonably straightforward linear equation to solve.

Upon solving we have:

    \begin{align*}-x+16&=3x-8 \\16&=4x-8 &\text{add x to both sides}\\24&=4x &\text{add 8 to both sides}\\6&=x &\text{divide both sides by 4}\\x&=6 &\text{swap sides}\end{align*}

We now replace the x with 6 on either original equation to find y. Using y=-x+16 we have y=-6+16=10.

Solution: x=6, y=10.

On the first applet, take the second expression for y and substitute it into the first equation, to find if there is an x coordinate that yields the same y coordinate on both lines. Remember that for coincident lines, there are infinitely many solutions; and for parallel lines, there are is no solution.

 

 

Substitute into the middle of an expression (Skill #4)

On the second applet, one of the equations is given in general form. Take the expression for y in the second equation and substitute it into the first to find the solution x that yields the same y on both lines.

Example:

    \[\begin{cases}3x+2y-30=0 \\ y=4x-18 \end{cases}\]

    \begin{align*}3x+2(4x-18)-30&=0 &\text{substitute }y=4x-18 \\ 3x+8x-36-30&=0 &\text{expand brackets}\\ 11x-66&=0 &\text{simplify}\\11x&=66 &\text{add 66} \\x&=6 &\text{divide by 11} \\y&=4(6)-18 = 24-18=6 &\text{substitute x into one original equation to find y} \end{align*}

Solution: x=6, y=6.

Substituting when neither variable is the subject of either equation (Skill #4)

Rearrange one or the other of the equations to make either x or y the subject. Then substitute into the other equation to find one part of the solution. Complete by finding the second part. Enter the values into the x, y input boxes to check the solution and to see the graphical representation of the solution.