mr barton maths

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Simultaneous Equations

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Foundational skills
Recognise simultaneous equations
\[ \text{Which is a pair?} \]
Identify when two equations form a pair.
Verify a solution
\[ x = 2, y = 3 \text{ satisfy both?} \]
Check if values make both equations true.
Identify matching coefficients
\[ 2x + 3y, \; 2x – y \]
Spot when coefficients match.
Decide: add or subtract
\[ +3y \text{ and } -3y \rightarrow \text{add} \]
Determine operation to eliminate variable.
Decide which variable to eliminate
\[ \text{Easier: } x \text{ or } y\text{?} \]
Identify which is easier to eliminate.
Elimination – same coefficients
Add equations to eliminate
\[ (+y) + (-y) = 0 \]
Add when coefficients are opposites.
Subtract equations to eliminate
\[ 2y – 2y = 0 \]
Subtract when coefficients are the same.
Solve by elimination (addition)
\[ 2x + y = 7, \; 5x – y = 14 \]
Full solution using addition.
Solve by elimination (subtraction)
\[ 3x + 2y = 13, \; x + 2y = 9 \]
Full solution using subtraction.
Elimination – scale one equation
Identify the required multiplier
\[ x \text{ and } 3x \rightarrow \times 3 \]
Find what to multiply by.
Multiply an equation by a constant
\[ 2x + 3y = 7 \xrightarrow{\times 4} \text{?} \]
Scale every term in an equation.
Solve after scaling one equation
\[ x + 2y = 8, \; 3x – y = 3 \]
Scale one equation then eliminate.
Elimination – scale both equations
Find multipliers using LCM
\[ 2y, 3y \rightarrow \times 3, \times 2 \]
Find multipliers for both equations.
Solve after scaling both equations
\[ 2x + 3y, \; 3x + 2y \]
Scale both equations then eliminate.
Elimination – rearrangement needed
Rearrange to standard form
\[ 2x = 3y + 7 \rightarrow ax + by = c \]
Rewrite into standard form.
Solve equations requiring rearrangement
\[ 2x + y = 10, \; y = x + 1 \]
Rearrange then solve.
Substitution method
Substitute an expression
\[ y = 2x + 1 \text{ into } 3x + y = 10 \]
Replace y to get equation in x only.
Solve by substitution (y isolated)
\[ y = 3x – 2, \; 2x + y = 8 \]
Solve when y is already isolated.
Solve by substitution (x isolated)
\[ x = 2y + 1, \; 3x – 2y = 7 \]
Solve when x is already isolated.
Rearrange then substitute
\[ x + y = 7, \; 2x + 3y = 17 \]
Rearrange one equation then substitute.
Forming equations
Form equations from numbers
\[ \text{Sum is 15, difference is 3} \]
Write equations from number context.
Form equations from costs
\[ \text{3 coffees + 2 teas = £9.50} \]
Write equations from cost context.
Form equations from geometry
\[ \text{Perimeter } = 26, \; l = w + 3 \]
Write equations from geometry context.
Form and solve from word problem
\[ \text{Find both numbers} \]
Form equations and solve them.
Graphical methods
Read the solution from a graph
\[ \text{Find intersection point} \]
Find x and y where lines cross.
Estimate a non-integer solution
\[ x \approx 1.5, \; y \approx 2.5 \]
Estimate when lines meet between grid points.
Find intersection point algebraically
\[ y = 2x + 1, \; y = -x + 7 \]
Calculate where two lines meet.
Non-linear (Higher)
Substitute linear into y = x²
\[ y = x + 2, \; y = x^2 \]
Find where line meets parabola.
Substitute linear into quadratic
\[ y = 2x – 1, \; y = x^2 – 2x + 3 \]
Find intersection with general quadratic.
Line and circle intersection
\[ y = x + 1, \; x^2 + y^2 = 5 \]
Find where line crosses circle.
Solve linear/quadratic (rearrange)
\[ x + y = 3, \; y = x^2 + 1 \]
Rearrange linear before substituting.
Interpret number of solutions
\[ \text{0, 1, or 2 intersections?} \]
Use discriminant to determine count.
Special cases
Equations with no solution
\[ y = 2x + 3, \; y = 2x – 1 \]
Recognise parallel lines.
Equations with infinitely many solutions
\[ 2x + 4y = 10, \; x + 2y = 5 \]
Recognise same line.
Solve with fractional coefficients
\[ \frac{1}{2}x + y = 4 \]
Solve equations containing fractions.
One variable equals zero
\[ x = 0 \text{ or } y = 0 \]
Handle zero or negative solutions.
Timer (Optional)
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