Solve simultaneous equations by substitution

Equations

\(\displaystyle \begin{gathered}y=2x+4\\5x-y=11\end{gathered}\)

Which expression can immediately replace y in the second equation?

Keep y equal to twice x plus four. Predict how five times x minus y changes when x increases from 3 to 4

A residual is the left side minus the right side of an equation; zero means it holds. Change the trial x. The first equation determines y. Find where the second residual is zero, then check the neighbouring settings

Which pair makes both residuals zero?

Substitution preserves both conditions

Use one equation to express a variable, then impose the other condition on that expression

1. 1Choose a useful subject Either equation may supply the expression. Prefer an isolated variable or a coefficient of 1 or -1 to avoid unnecessary fractions
2. 2Replace in the other equation Replace the complete variable by its equal expression. Keep brackets when a coefficient or a minus sign acts on it \(\displaystyle 5x-(2x+4)=11\)
3. 3Solve and substitute back Find one unknown, then use an original equation to find the other. A simultaneous solution needs both values
4. 4Check both originals Check each original equation independently. Substitution into the source equation alone gives an identity and no new information
5. 5Recognise cancellation A contradiction means no solution. An identity means equivalent line equations and infinitely many pairs, not that every pair in the plane works

Equations

\(\displaystyle \begin{gathered}y=2x+4\\5x-y=11\end{gathered}\)

Which single-variable equation correctly replaces y?

Equations

\(\displaystyle 5x-(2x+4)=11\)

Solve the substituted equation for x.

Equations

\(\displaystyle egin{gathered}y=2x+4\5x-y=11end{gathered}\)

Use x=5 to find y.

Original pair

\(\displaystyle \begin{gathered}y=3x-2\\6x-y=17\end{gathered}\)

Which line is the first error?

Equations

\(\displaystyle egin{gathered}y=2x+4\5x-y=11end{gathered}\)

A student substitutes y=2x+4 into that same source equation and obtains 2x+4=2x+4. What should they do next?

Equations

\(\displaystyle \begin{gathered}2x+y=7\\3x-2y=0\end{gathered}\)

Make x the subject of the first equation so it can be substituted into the second. Which rearrangement is correct?

Equations

\(\displaystyle 3\left(\frac{7-y}{2}\right)-2y=0\)

Substitute x=(7−y)/2 into the other equation and find y.

Equations

\(\displaystyle \begin{gathered}2x+y=7\\3x-2y=0\end{gathered}\)

Use y=3 to find x and check the original pair

Equations

\(\displaystyle \begin{gathered}y=2x+1\\3x+y=4\end{gathered}\)

Find x exactly. Do not round during substitution

Equations

\(\displaystyle egin{gathered}y=2x+1\3x+y=4end{gathered}\)

Find y using three fifths as the value of x

Equations

\(\displaystyle \begin{gathered}x=-2\\3x+2y=8\end{gathered}\)

A constant is also a valid subject expression. Find y.

Equations

\(\displaystyle \begin{gathered}2x-y=5\\y=3\end{gathered}\)

The second equation gives a constant subject. Find x.

Equations

\(\displaystyle \begin{gathered}y=2x+1\\4x-2y=5\end{gathered}\)

Substitution gives −2=5. What does that tell you?

Equations

\(\displaystyle \begin{gathered}y=2x+1\\4x-2y=-2\end{gathered}\)

Substitution gives −2=−2. What is the solution set?

Use an exact check

A table can locate a candidate. Algebra can establish its exact value. Check the final pair in both original equations, then apply any restrictions from the situation

1. 1Use an equivalent expression Every substitution must preserve equality; do not replace only part of an expression
2. 2Keep fractions exact Round only when the question requests it, after checking the exact pair
3. 3Report both meanings A context may ask for one quantity, but the other still needs to satisfy its condition. Whole-item counts must be non-negative integers