Solve a linear equation

Friends sharing the bill

\(\displaystyle 5\)

Voucher

\(\displaystyle \$12\)

Each friend paid

\(\displaystyle \$23\)

Five friends used a $12 voucher on a restaurant bill and split the rest equally. Each friend paid $23. What was the bill before the voucher was used?

The tiles represent k(x - 3): k x-tiles and 3k hollow unit tiles. Hollow tiles are subtracted units. Move k from 1 to 5 and count both kinds of tile, watching the readouts

As k changed, for every position of the slider, what happened to the units taken away?

Solving linear equations

A linear equation is solved by undoing the operations around the pronumeral in reverse order, doing the same to both sides. A factor outside a bracket multiplies every term inside. In the formula below, a must be non-zero

1. 1Undo operations in reverse order The division by 5 happened last, so it is undone first by multiplying both sides by 5 \(\displaystyle \begin{gathered}\frac{B-12}{5}=23\\\Rightarrow B-12=115\end{gathered}\)
2. 2Keep both sides balanced Each undo step is applied to both sides, so the equation stays balanced until the bill stands alone \(\displaystyle \begin{gathered}B-12+12=115+12\\\Rightarrow B=127\end{gathered}\)
3. 3Expand every bracket term A factor outside a bracket multiplies every term inside, including any subtracted constant \(\displaystyle \begin{gathered}5(x-3)=35\\\Rightarrow 5x-15=35\end{gathered}\)
4. 4Collect like terms Move the unknown terms to one side and the numbers to the other side, then divide by the remaining coefficient \(\displaystyle \begin{gathered}18+4x=7x\\\Rightarrow 18=3x\\\Rightarrow x=6\end{gathered}\)

Plan A joining fee

\(\displaystyle \$60\)

Plan A monthly cost

\(\displaystyle \$45\)

Plan B monthly cost

\(\displaystyle \$57\)

Plan C joining fee

\(\displaystyle \$120\)

Plan C monthly cost

\(\displaystyle \$42\)

Three gym plans charge as follows. Plan A costs $60 to join plus $45 a month. Plan B has no joining fee and costs $57 a month. Plan C costs $120 to join plus $42 a month. After how many months have Plans A and B cost the same in total?

Write each plan's total after m months, then set the two totals equal

Plan A joining fee

\(\displaystyle \$60\)

Plan A monthly cost

\(\displaystyle \$45\)

Plan B monthly cost

\(\displaystyle \$57\)

Plan C joining fee

\(\displaystyle \$120\)

Plan C monthly cost

\(\displaystyle \$42\)

Plan B has no joining fee. After how many months have Plans C and B cost the same in total?

Write each plan's total after m months, then set the two totals equal and collect like terms

Which line correctly expands the bracket?