Year 10AC9M10A02

Simultaneous Equations Algebraically

Solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation.

Two unknowns need two equations. Eliminate one variable, solve for the other, then substitute back.

Builds onYr 10 · Simultaneous Equations from a Graph

Method

Substitution suits an equation already solved for a variable

y = \ldots

Elimination suits matching coefficients

Add or subtract to remove one variable.

Multiply a whole equation to make them match

Every term, or the equation changes.

Always substitute back to find the second variable

And check in the equation you did not use.

Worked example

e.g.

\text{Solve } 3x + 2y = 16 \text{ and } 5x - 2y = 8

Look at the coefficients of $y$ . They are $+2$ and $-2$ . $\text{already opposite}$
Add the equations to eliminate $y$ . $8x = 24$
Solve for $x$ . $x = 3$
Substitute back into the first equation. $3(3) + 2y = 16$
Solve for $y$ . $2y = 7 \ \Rightarrow\ y = 3{,}5$
Check in the other equation. $5(3) - 2(3{,}5) = 8 \quad\checkmark$

Practice

Eliminate, solve, substitute back, then check in the unused equation.

x + y = 10,\ x - y = 4

Answer

x = 7,\ y = 3

2x + y = 11,\ x + y = 7

Answer

x = 4,\ y = 3

3x + 4y = 18,\ x = 2y

Answer

x = 4,\ y = 2

\text{Why check in the other equation?}

Answer

\text{The one you used will work even if you slipped earlier.}

\text{Two lines are parallel. How many solutions?}

Answer

\text{None.}

\text{When is substitution easier than elimination?}

Answer

\text{When one equation is already solved for a variable.}