Simultaneous Equations

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Notes

Linear Simultaneous Equations

**Linear simultaneous equations** involve two unknowns (usually x and y) and two equations, e.g. $3x + 2y = 11$ and $2x - y = 5$ .
The solution is the pair of values (x, y) that satisfy **both** equations simultaneously.
Solve by **elimination**: make the coefficients of one variable the same, then add or subtract to eliminate it.
If the signs in front of the term to eliminate are **the same**, **subtract** the equations.
If the signs are **different**, **add** the equations.
Solve the resulting equation for one variable, then substitute back to find the other.
Solve by **substitution**: rearrange one equation to $y =$ ... $(or x = ...)$ and substitute into the other equation.
Always **check** your final solutions by substituting into both original equations.

Graphical Solution of Linear Simultaneous Equations

Plot both equations on the same set of axes (use a table of values or rearrange to $y = mx + c)$ .
The **point of intersection** gives the solution: x-coordinate is the x-value, y-coordinate is the y-value.
If the lines are parallel, there is **no solution** (inconsistent equations).
If the lines are the same, there are **infinitely many solutions** (dependent equations).

Forming Simultaneous Equations from Context

Introduce letters (e.g. x, y) to represent the unknowns, with clear units.
Write two equations based on the given information (e.g. costs, totals, ratios).
Solve the equations simultaneously, then **answer the question** in context (e.g. cost of an apple is 40p).
Sometimes you need to find another quantity (e.g. product xy) after solving.

Quadratic Simultaneous Equations

Involve one **linear** equation and one **quadratic** (or non-linear) equation (e.g. $x^{2} + y^{2} = 25$ and $y - 2x = 5)$ .
Solve by **substitution**: rearrange the linear equation to $y =$ ... $(or x = ...)$ and substitute into the quadratic.
Expand and solve the resulting quadratic equation (may factorise, use formula, or complete square).
Substitute each x-value back into the linear equation to find the corresponding y-value.
Present solutions as **pairs** (e.g. $x = 0, y = 5$ or $x = -4, y = -3)$ .
If the quadratic has a **repeated root**, the line is **tangent** to the curve (one intersection).
If the quadratic has **no real roots**, the line and curve **do not intersect** (no solutions).

Special Cases in Quadratic Simultaneous Equations

If the linear equation is not in the form $y =$ ... or $x =$ ..., **rearrange it first** before substituting.
For equations like $xy = 3$ and $x + y = 4$ , rearrange either to $y = 4 - x$ or $y = \frac{3}{x}$ and substitute.
Be careful with fractions: multiply through to clear denominators when necessary.
Always check solutions satisfy **both** original equations.

Graphical solution of linear simultaneous equations

Graphical solution of quadratic simultaneous equations

Elimination method step-by-step

Substitution method for quadratic simultaneous equations

Practice questions

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1.For linear simultaneous equations, which method involves making the coefficients of one variable the same in both equations and then adding or subtracting to eliminate that variable?
Easy
- AElimination method
- BSubstitution method
- CGraphical method
- DTrial and error method
2.When solving linear simultaneous equations graphically, the solution is given by the coordinates of the point where the lines:
Easy
- AIntersect
- BAre parallel
- CAre coincident
- DCross the x-axis
3.To solve the simultaneous equations $3x + 2y = 11$ and $2x - y = 5$ by elimination, you could multiply the second equation by 2 and then:
Easy
- AAdd to the first equation
- BSubtract from the first equation
- CMultiply by the first equation
- DDivide by the first equation
4.Solve the simultaneous equations: $2x + y = 7, 3x - y = 8$ . What is the value of x?
Medium
- A3
- B2
- C1
- D4
5.Solve the simultaneous equations: $3x - 8y = 22, x + 4y = 4$ . What is the value of y?
Medium
- A-0.5
- B-1
- C1
- D2
6.3 apples and 5 bananas cost £1.80. 5 apples and 1 banana cost £2.30. If a is the price of an apple in pence and b is the price of a banana in pence, which pair of equations represents this situation?
Medium
- A $3a + 5b = 180, 5a + b = 230$
- B $3a + 5b = 1.80, 5a + b = 2.30$
- C $3a + 5b = 180, 5a + b = 2.30$
- D $3a + 5b = 1.80, 5a + b = 230$
7.Solve the simultaneous equations: $5x + 8y = 4, (\frac{1}{2})x + 3y = 7$ . What is the value of x?
Medium
- A-4
- B4
- C20
- D-20
8.Solve algebraically the simultaneous equations: $x^{2} + y^{2} = 25, y - 2x = 5$ . Which of the following is a solution?
Hard
- A $x = 0, y = 5$
- B $x = 0, y = -5$
- C $x = 4, y = 3$
- D $x = -4, y = 3$

Summary

## Simultaneous Equations - Revision Notes ### Linear Simultaneous Equations - **What they are:** Two equations with two unknowns (x, y) that are both linear (no x², y², xy). - **Solving by Elimination:** 1. Make the coefficients of one variable the same in both equations (multiply if needed). 2. If signs are the same, **subtract** equations; if different, **add** equations to eliminate that variable. 3. Solve the resulting equation for the remaining variable. 4. Substitute back into one original equation to find the other variable. 5. Check both solutions satisfy the original equations. - **Solving by Substitution:** 1. Rearrange one equation to y = ... or x = ... 2. Substitute this expression into the other equation. 3. Solve for the variable, then substitute back to find the other. - **Solving Graphically:** Plot both lines; the intersection point gives the solution (x, y). - **Forming Equations:** Define variables (e.g., x = price of apple, y = price of banana) and write two equations from the problem context. Solve and answer in context with units. ### Quadratic Simultaneous Equations - **What they are:** One linear and one quadratic (e.g., x² + y² = 25) equation. - **Solving by Substitution (algebraic method):** 1. Rearrange the linear equation to y = ... or x = ... 2. Substitute into the quadratic equation. 3. Expand and solve the resulting quadratic (factorise or use formula). 4. Substitute each x (or y) back into the linear equation to find corresponding y (or x). 5. Present solutions as pairs (e.g., x = 0, y = 5 or x = -4, y = -3). - **Graphical Interpretation:** The intersection(s) of the line and curve give the solutions. A repeated root means the line is tangent; no real roots means no intersection. ### Key Points - Always check solutions satisfy **both** original equations. - Write final answers as clear pairs (x = ..., y = ...). - For word problems, answer the question asked (e.g., total cost, not just x and y).

Key terms

Simultaneous Equations: Two or more equations that share variables; the solution satisfies all equations at once.
Elimination Method: A method to solve simultaneous equations by adding or subtracting the equations to remove one variable.
Substitution Method: A method to solve simultaneous equations by rearranging one equation and substituting it into the other.
Quadratic Simultaneous Equations: A pair of equations where at least one is quadratic (contains x², y², or xy).
Coefficient: The number multiplying a variable (e.g., in 3x, the coefficient is 3).
Point of Intersection: The point where two graphs meet; its coordinates are the solution to the simultaneous equations.

Slides

What are Simultaneous Equations?

Simultaneous equations involve two unknowns (usually x and y) and two equations.
They are called linear if there are no $x^{2}, y^{2}$ , or xy terms.
The solution is the pair (x, y) that satisfies both equations at once.

Which of the following is a pair of linear simultaneous equations?

$y = 2x + 1$ and $y = x^{2} + 3$
$3x + 2y = 7$ and $x - y = 4$
$x^{2} + y^{2} = 25$ and $y = 2x$
$xy = 6$ and $x + y = 5$

Solving by Elimination

Elimination removes one variable by making its coefficients the same.
If the signs are the same, subtract the equations; if different, add them.
Solve the resulting equation for one variable, then substitute back to find the other.

To eliminate y from $3x + 2y = 11$ and $2x - y = 5$ , multiply the second equation $by ____ and$ then $____ the$ equations.

Solving by Substitution

Rearrange one equation to make x or y the subject.
Substitute this expression into the other equation.
Solve the resulting equation, then substitute back to find the other variable.

To solve $3x + 2y = 11$ and $2x - y = 5$ by substitution, which rearrangement is correct?

$y = 2x - 5$
$y = 5 - 2x$
$x = 2y + 5$
$x = 5 - 2y$

Solving Graphically

Plot both equations on the same axes.
The point(s) of intersection give the solution(s).
This method works for both linear and quadratic simultaneous equations.

Which of these are true about solving simultaneous equations?

Elimination and substitution are algebraic methods.
Graphical method always gives exact solutions.
You can check your answer by substituting into the original equations.

Forming Simultaneous Equations from Context

Introduce two letters (e.g., x and y) for the unknowns.
Write two equations based on the given information.
Solve the equations and interpret the answer in context.

3 apples and 2 bananas cost £1.80. 5 apples and 1 banana cost £2.30. If $x =$ price of an apple in pence, the equations are $3x + 2y = ____ and 5x + y = ____$ .

Quadratic Simultaneous Equations

At least one equation is quadratic $(contains x^{2}, y^{2}$ , or xy).
Use substitution: rearrange the linear equation into $y =$ ... or $x =$ ...
Substitute into the quadratic equation and solve the resulting quadratic.

To solve $x^{2} + y^{2} = 25$ and $y - 2x = 5$ , what should you substitute?

$y = 2x + 5$ into $x^{2} + y^{2} = 25$
$x^{2} + y^{2} = 25$ into $y = 2x + 5$
$y = 5 - 2x$ into $x^{2} + y^{2} = 25$
$x = (y - 5)/2$ into $x^{2} + y^{2} = 25$

Special Cases in Quadratic Simultaneous Equations

If the resulting quadratic has a repeated root, the line is tangent to the curve (one solution).
If the quadratic has no real roots, the line does not intersect the curve (no solutions).
Always check your solutions by substituting back into both original equations.

Which statements are true?

A repeated root means the line is tangent to the curve.
No real roots means the line intersects the curve twice.
You should always check your solutions.

Worked Example: Linear

Solve $5x + 2y = 11$ and $4x - 3y = 18$ .
Eliminate y: multiply first by 3, second by 2 → $15x + 6y = 33$ and $8x - 6y = 36$ .
Add: $23x = 69$ → $x = 3$ . Substitute to find $y = -2$ .

In the worked example, after eliminating y, the equation becomes $23x = ____$ . So $x = ____$ .

Worked Example: Quadratic

Solve $x^{2} + y^{2} = 36$ and $x - 2y = 6$ .
Rearrange linear: $x = 2y + 6$ .
Substitute: $(2y + 6)^{2} + y^{2} = 36$ → $5y^{2} + 24y = 0$ → $y = 0$ or $y = -\frac{24}{5}$ .
Corresponding $x: x = 6$ or $x = -\frac{18}{5}$ .

For $x^{2} + y^{2} = 36$ and $x - 2y = 6$ , what are the solutions?

$x = 6, y = 0$ and $x = -\frac{18}{5}, y = -\frac{24}{5}$
$x = 6, y = 0$ only
$x = 0, y = 6$ and $x = -\frac{24}{5}, y = -\frac{18}{5}$
$x = -6, y = 0$ and $x = \frac{18}{5}, y = \frac{24}{5}$

Key Tips

Always check your solutions in both original equations.
Write solutions as clear pairs (x, y).
In context questions, answer with appropriate units.

Which are good practices?

Check your answer by substituting back.
Only write the x value; y is not needed.
In word problems, include units in your answer.

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