Solve a pair of simultaneous linear equations

First equation

\(\displaystyle y=3x+2\)

Second equation

\(\displaystyle x+2y=18\)

Which equation correctly replaces the second unknown?

Keep the replacement expression in brackets before expanding

Rule A output

\(\displaystyle A=2t+3\)

Rule B output

\(\displaystyle B=15-2t\)

Shared input

\(\displaystyle t\)

Both rules use the same input. Each bar shows the output of its rule. Find where the bars have the same height

At the matching position, what are the input and the shared output, in that order?

Substitute the whole expression

A simultaneous solution makes both equations true. Use an equation with an unknown already isolated to replace that unknown in the other equation. Then solve the equation with one unknown

1. 1Replace the isolated unknown Keep the replacement in brackets because the second equation multiplies it \(\displaystyle x+2(2x+4)=23\)
2. 2Expand and collect Multiply every term inside the bracket, then collect like terms \(\displaystyle \begin{aligned}x+4x+8&=23\\5x+8&=23\end{aligned}\)
3. 3Solve for the first unknown Subtract the constant from both sides, then divide by the coefficient \(\displaystyle \begin{aligned}5x&=15\\x&=3\end{aligned}\)
4. 4Find the other unknown Substitute the value into the equation with the isolated unknown \(\displaystyle \begin{aligned}y&=2(3)+4\\y&=10\end{aligned}\)
5. 5Check both original equations Both equations must be true for the same pair \(\displaystyle \begin{gathered}10=2(3)+4\\3+2(10)=23\\(x,y)=(3,10)\end{gathered}\)
6. 6When both equations have the same subject Set the two expressions equal. This is also substitution. Then solve and find the other unknown as before \(\displaystyle \begin{gathered}y=2x+5\\y=11-x\\2x+5=11-x\end{gathered}\)

First equation

\(\displaystyle y=3x-2\)

Second equation

\(\displaystyle 2x+y=23\)

Find the first unknown in this simultaneous pair

Replace the second unknown in the second equation with the whole expression from the first equation

First equation

\(\displaystyle y=9+\frac{x}{2}\)

Second equation

\(\displaystyle y=1-\frac{3x}{2}\)

Find the second unknown in this simultaneous pair

Both equations already have the same subject. Equate the two expressions before finding the requested value

First equation

\(\displaystyle y=4x-3\)

Second equation

\(\displaystyle 2x-y=7\)

Which line contains the first error?

Check how each term changes when the bracket is removed