Mathematics » Equations and Inequalities » Solving Simultaneous Equations

Solving by Substitution

Solving by Substitution

  • Use the simplest of the two given equations to express one of the variables in terms of the other.

  • Substitute into the second equation. By doing this we reduce the number of equations and the number of variables by one.

  • We now have one equation with one unknown variable which can be solved.

  • Use the solution to substitute back into the first equation to find the value of the other unknown variable.

The following video shows how to solve simultaneous equations using substitution.

Example

Question

Solve for \(x\) and \(y\):

\begin{align*} x – y & =1 \qquad \ldots(1) \\ 3 & = y – 2x \qquad \ldots(2) \end{align*}

Use equation \((1)\) to express \(x\) in terms of \(y\)

\[x=y+1\]

Substitute \(x\) into equation \((2)\) and solve for \(y\)

\begin{align*} 3 & = y – 2(y + 1) \\ 3 & = y – 2y – 2 \\ 5 & = -y \\ \therefore y & = -5 \end{align*}

Substitute \(y\) back into equation \((1)\) and solve for \(x\)

\begin{align*} x & = (-5) + 1 \\ \therefore x & = -4 \end{align*}

Check the solution by substituting the answers back into both original equations

Write the final answer

\begin{align*} x & = -4 \\ y & = -5 \end{align*}

Example

Question

Solve the following system of equations:

\begin{align*} 4y + 3x & = 100 \qquad \ldots(1) \\ 4y – 19x & = 12 \qquad \ldots (2) \end{align*}

Use either equation to express \(x\) in terms of \(y\)

\begin{align*} 4y + 3x & = 100 \\ 3x & = 100 – 4y \\ x & = \cfrac{100 – 4y}{3} \end{align*}

Substitute \(x\) into equation \((2)\) and solve for \(y\)

\begin{align*} 4y – 19\left(\cfrac{100 – 4y}{3}\right) & = 12 \\ 12y – 19(100 – 4y) & = 36 \\ 12y – \text{1 900} + 76y & = 36 \\ 88y & = \text{1 936} \\ \therefore y & =22 \end{align*}

Substitute \(y\) back into equation \((1)\) and solve for \(x\)

\begin{align*} x & = \cfrac{100 – 4(22)}{3} \\ & = \cfrac{100 – 88}{3} \\ & = \cfrac{12}{3} \\ \therefore x & = 4 \end{align*}

Check the solution by substituting the answers back into both original equations

Write the final answer

\begin{align*} x & = 4 \\ y & = 22 \end{align*}

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