Solve by substitution
Find a pair of values that makes both equations true. Replacing one unknown with an equal expression leaves an equation with only one unknown
- Choose the ready subject Use the equation that already has an unknown isolated. Either unknown can be the subject
- Replace the whole expression Insert the expression into the other equation. Bracket it when it is multiplied or subtracted
- Solve, then substitute back Solve the resulting equation. Use that value to find the remaining unknown
- Use both original equations Check the same pair in both. A pair that works in only one equation is not a simultaneous solution
- Same subject in both equations Equate the expressions for that subject, then solve and substitute back
Worked example
Solve the simultaneous pair
\[\begin{gathered}y=6+\frac{x}{2}\\y=2-\frac{3x}{2}\end{gathered}\]
- \(\displaystyle 6+\frac{x}{2}=2-\frac{3x}{2}\)Equate the expressions for the shared unknown
- \(\displaystyle 12+x=4-3x\)Multiply every term on both sides by the common denominator
- \(\displaystyle 12+4x=4\)Add the variable term from the right to both sides
- \(\displaystyle 4x=-8\)Subtract the constant on the left from both sides
- \(\displaystyle x=-2\)Divide both sides by the coefficient
- \(\displaystyle y=6+\frac{-2}{2}\)Substitute into the first original equation
- \(\displaystyle y=5\)Calculate the other unknown
- \(\displaystyle 6+\frac{-2}{2}=5\)The first original equation gives the required value
- \(\displaystyle 2-\frac{3(-2)}{2}=5\)The second original equation gives the same value
- \(\displaystyle (x,y)=(-2,5)\)State the two values in the named order
Common mistakes
Multiplying only part of the replacement
✕\(\displaystyle 2(3x+4)=6x+4\)✓\(\displaystyle 2(3x+4)=6x+8\)
The outside coefficient multiplies every term in the bracket. Keep the bracket until you have distributed the coefficientLosing a sign when subtracting
✕\(\displaystyle -(2x-5)=-2x-5\)✓\(\displaystyle -(2x-5)=-2x+5\)
Subtract the entire replacement. Both terms change sign when the bracket is removedChecking only one equation
✕\(\displaystyle \begin{gathered}y=x+2\\x+y=12\\(x,y)=(3,5)\end{gathered}\)✓\(\displaystyle \begin{gathered}(x,y)=(5,7)\\7=5+2\\5+7=12\end{gathered}\)
The proposed pair fits the first equation but its sum does not fit the second. A simultaneous pair must satisfy both originalsDifficulty levels
introSubstitute an expression from one equation into the other. Find an integer pair
standardEquate two expressions for the same unknown. Work with signed values and simple fractional coefficients
Questions about this topic
Must both equations have an unknown isolated?+
No. One ready subject is enough. Replace that unknown in the other equation with the whole expression
Why use brackets around the replacement?+
Brackets show that a coefficient or subtraction applies to every term in the expression
Why do I need to find both values?+
A simultaneous solution gives a value for each unknown. Check those same values in both original equations
What if both equations already have the same subject?+
Set their expressions equal. They must have the same value at the solution
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 2: Linear equations
In this chapter
Translate between a description and an algebraic expressionEvaluate an algebraic expression by substitutionTranslate between a description and an algebraic expressionSimplify algebraic terms and collect like termsSimplify algebraic terms and collect like termsExpand brackets and collect like terms2D Two-step linear equations: Solve a linear equation2E Linear equations with brackets: Solve a linear equationSet up and solve a practical linear equationSolve and interpret a linear inequationRepresent and read inequalities on a number lineSubstitute values into a formulaSolve for an unknown or recognise a rearranged formulaSolve a pair of simultaneous linear equations (optional extension)Solve a pair of simultaneous linear equations by elimination (optional extension)Set up a pair of simultaneous equations from a description (optional extension)Solve a practical problem with simultaneous equations (optional extension)