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Simultaneous Equations

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Foundational skills
Substitute to find the other unknown
\[ x=5:\ 3x+2y=19 \]
Put a known value in, then solve for the other letter.
Add or subtract to eliminate?
\[ \text{Add or subtract? } 3y,\ -3y \]
Decide whether adding or subtracting cancels a letter.
Subtract one equation from another
\[ (5x+2y)-(5x-3y) \]
Subtract every term, watching out for negatives.
Multiply an equation through
\[ \times 3:\ 2x-y=5 \]
Multiply every term, including the number on the right.
Find the smallest multipliers
\[ 2y,\ 3y \to \times 3,\ \times 2 \]
Find what to multiply each equation by so the terms match.
Is this pair a solution of both?
\[ x=2,\ y=-1\text{ in both?} \]
Check whether a pair of values works in both equations.
Elimination
Same sign, all positive, so subtract
\[ 3x+2y=16,\ x+2y=8 \]
Matching terms with the same sign: subtract the equations.
Opposite signs, so add
\[ 4x+3y=18,\ 2x-3y=0 \]
Matching terms with different signs: add the equations.
Both negative, so subtract
\[ 5x-2y=11,\ x-2y=-1 \]
Both matching terms are negative: subtract carefully.
Scale one equation, then subtract
\[ 2x+y=7,\ 3x+2y=11 \]
Multiply one equation so the terms match, then subtract.
Scale one equation, then add
\[ 3x-y=5,\ 2x+3y=18 \]
Multiply one equation so the terms match, then add.
Scale both equations
\[ 3x+2y=12,\ 2x+5y=19 \]
Multiply both equations to make matching terms.
Scale both, with negative terms
\[ 4x-3y=-1,\ 3x+2y=12 \]
Scale both equations when some terms are negative.
Rearrange first
Terms on the wrong side
\[ 2x=11-3y,\ x+y=4 \]
Rearrange so the letters line up, then eliminate.
Terms in a different order
\[ y+3x=10,\ 2x-y=5 \]
Line up x with x and y with y before eliminating.
Clear fractions first
\[ \tfrac{1}{2}x+y=4,\ x-y=5 \]
Multiply through to remove fractions, then eliminate.
Substitution
Substitute y = …
\[ y=2x-1,\ 5x-y=10 \]
Replace y in the other equation, using a bracket.
Substitute x = …
\[ x=3y+2,\ 2x+y=18 \]
Replace x in the other equation, using a bracket.
Both have y as the subject
\[ y=3x-2,\ y=x+6 \]
Set the two expressions for y equal to each other.
Rearrange, then substitute
\[ x+y=7,\ 2x+3y=17 \]
Make one letter the subject, then substitute it.
Graphical
Read the solution from a graph
\[ \text{Solve using the graph} \]
The solution is where the two lines cross.
Where do two lines meet?
\[ y=2x+1,\ x+y=7 \]
Solve the equations to find the crossing point.
How many solutions?
\[ y=2x+3,\ y=2x-1 \]
Decide between no solutions, one, or infinitely many.
Non-linear (Higher)
Line meets y = x²
\[ y=x+2,\ y=x^2 \]
Find where a straight line crosses the curve y = x².
Line meets a quadratic
\[ y=2x-1,\ y=x^2-2x+2 \]
Find where a straight line crosses a quadratic curve.
Rearrange the line, then substitute
\[ x+y=3,\ y=x^2+1 \]
Make y the subject of the straight line first.
Line and circle
\[ y=x+1,\ x^2+y^2=5 \]
Find where a straight line crosses a circle.
How many intersections?
\[ y=x^2+3,\ y=2x+2 \]
Use the discriminant to count crossing points: 0, 1 or 2.
Extensions
Two numbers: sum and difference
\[ x+y=23,\ x-y=7 \]
Form two equations from the facts, then solve.
Shopping: cost of one item
\[ 3p+2r=140,\ p+2r=80 \]
Form and solve equations to find the cost of one item.
Solutions are fractions
\[ 4x+2y=7,\ 2x-2y=1 \]
Solve when the answers are not whole numbers.
Form equations from a shape
\[ \text{Opposite sides are equal} \]
Opposite sides of a rectangle are equal: form and solve.
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