Simultaneous equations

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What simultaneous equations are

A simultaneous equations problem is a pair of linear equations that involve the same two variables, usually x and y.
A linear equation contains variables only to the first power, with no x², xy or 1/x terms.
A solution is a pair of values (x, y) that makes both equations true at the same time.
Example: the pair 2x + 3y = 6 and 4x + 9y = 15 has the solution x = 3/2, y = 1.
Because both equations must hold at once, the solution is written as an ordered pair, e.g. (3/2, 1).

Solving by substitution

Substitution means rearranging one equation to make one variable the subject, then putting that expression into the other equation.
Example: from 2x + 3y = 6, rearrange to get x = 3 − (3/2)y.
Substitute that expression into the second equation: 4(3 − (3/2)y) + 9y = 15.
This leaves a single equation in one variable, which you solve in the usual way.
In the example, solving gives y = 1; then substitute back to find x = 3/2.
Always substitute back into one of the original equations to check both values work.

Solving by elimination

Elimination means adding or subtracting the two equations to remove one variable.
If the coefficients of one variable are the same, subtract the equations; if they are opposites, add them.
If the coefficients do not match, multiply one or both equations by a number to make them match first.
Example: 2x + 3y = 6 and 4x + 9y = 15 — multiply the first equation by 2 to get 4x + 6y = 12.
Subtracting the second equation from this gives −3y = −3, so y = 1.
Substitute y = 1 back to find x = 3/2, giving the same solution as substitution.

Solving graphically

Each linear equation is a straight line when drawn on a set of axes.
The solution of a pair of simultaneous equations is the point where the two lines cross.
Read the coordinates of the crossing point to find x and y.
If the lines are parallel they never cross, so there is no solution.
If the two equations describe the same line, every point on the line is a solution.

Forming equations from word problems

Choose a letter for each unknown quantity, such as p for the price of one item and q for the price of another.
Translate each sentence of the problem into an equation using those letters.
Example: if 2 pens and 3 pencils cost £6, write 2p + 3q = 6.
Use a second piece of information to write the second equation, e.g. 4p + 9q = 15.
Solve the pair by elimination or substitution, then state the answer in the context of the problem.
Check the answer makes sense — for example, prices should be positive.

Checking your solution

Substitute your values of x and y into both original equations.
Both equations must give true statements, e.g. 2(3/2) + 3(1) = 6 and 4(3/2) + 9(1) = 15.
If one equation fails, the solution is wrong — go back and check your working.
A quick sketch of the two lines can confirm the crossing point is in a sensible place.

Key vocabulary

Variable: a letter standing for an unknown number, such as x or y.
Coefficient: the number multiplying a variable, e.g. the 2 in 2x.
Linear system: a collection of two or more linear equations in the same variables.
Solution: the values of the variables that satisfy every equation at once.
Ordered pair: the solution written as (x, y).

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1.In the pair of simultaneous equations 2x + 3y = 6 and 4x + 9y = 15, which variables are the unknowns?
Easy
- Ax and y
- B2 and 3
- C6 and 15
- D4 and 9
2.The solution of a pair of simultaneous linear equations is a pair of values that satisfies both equations at the same time.
Easy
True or false?
3.Which of these is a linear equation?
Medium
- A3x + 2y = 7
- Bx² + y = 5
- Cxy = 6
- Dx² + y² = 9
4.On a graph, the solution of two simultaneous linear equations is found at which point?
Medium
- AWhere the two straight lines cross
- BWhere each line meets the y-axis
- CThe origin
- DWhere each line meets the x-axis
5.The pair of equations 2x + 3y = 6 and 4x + 9y = 15 is solved by elimination. Which multiplier should be applied to the first equation so that the x terms cancel when the equations are added or subtracted?
Medium
- A2
- B3
- C4
- D9
6.Solve the simultaneous equations 2x + 3y = 6 and 4x + 9y = 15. What is the value of x?
Medium
- A3/2
- B1
- C3
- D6
7.Put the steps for solving 2x + 3y = 6 and 4x + 9y = 15 by substitution into the correct order.
Medium
- Substitute x = 3 - (3/2)y into 4x + 9y = 15 to get 4(3 - (3/2)y) + 9y = 15
- Expand and solve to find y = 1
- Substitute y = 1 back into x = 3 - (3/2)y to find x = 3/2
- Check that x = 3/2 and y = 1 satisfy both original equations
8.Which of the following pairs of values are solutions to the simultaneous equations 2x + 3y = 6 and 4x + 9y = 15? (select all that apply)
Hard
- Ax = 1.5, y = 1
- Bx = 3, y = 0
- Cx = 0, y = 2
- Dx = 3/2, y = 1
- Ex = 4, y = -1

Thêm câu hỏi

9.Which pair of simultaneous equations models this situation, where p is the cost of a pen in pounds and c is the cost of a pencil in pounds?
Medium
- A2p + 3c = 6 and 4p + 9c = 15
- B3p + 2c = 6 and 9p + 4c = 15
- C2p + 3c = 15 and 4p + 9c = 6
- D2p + 4c = 6 and 3p + 9c = 15
10.Which of the following are valid methods for solving a pair of simultaneous linear equations? (select all that apply)
Medium
- AElimination
- BSubstitution
- CGraphical method (finding the intersection point)
- DTrial and improvement
11.Match each equation with a point that lies on its line.
Medium
- y = 2x + 1
- y = 3x - 2
- y = -x + 4
- y = 0.5x + 3
- (1, 3)
- (2, 4)
- (3, 1)
- (0, 1)
12.Put the steps for solving the simultaneous equations 3x + 2y = 16 and 2x - y = 6 by elimination into the correct order.
Medium
- Multiply the second equation by 2 to get 4x - 2y = 12.
- Add the two equations to eliminate y, giving 7x = 28.
- Solve for x: x = 4.
- Substitute x = 4 into 2x - y = 6 to get 8 - y = 6, so y = 2.
- Check the solution (4, 2) in both original equations.
13.Solve the simultaneous equations 3x + 2y = 16 and 2x - y = 6. What is the value of x?
Medium
- Ax = 4
- Bx = 2
- Cx = 5
- Dx = 3
14.Solve the simultaneous equations x + y = 5 and 2x - y = 1. What is the value of x?
Medium
- Ax = 2
- Bx = 3
- Cx = 1
- Dx = 4
15.Solve the simultaneous equations 3x + 2y = 16 and 2x - y = 6. What is the value of x + y?
Hard
- A6
- B8
- C10
- D4
16.A student solved 3x + 2y = 16 and 2x - y = 6 by substitution. They rearranged the second equation as y = 2x - 6, then substituted into the first to get 3x + 2(2x - 6) = 16. Which equation results after simplifying?
Hard
- A7x - 12 = 16
- B7x + 12 = 16
- Cx - 12 = 16
- D7x = 4
17.Which of the following points lie on the line y = 2x + 1? (select all that apply)
Medium
- A(0, 1)
- B(1, 3)
- C(2, 5)
- D(3, 6)
- E(-1, -1)
18.The lines y = 2x + 1 and y = -x + 4 intersect at a point. What are the coordinates of this intersection?
Medium
- A(1, 3)
- B(2, 5)
- C(0, 4)
- D(3, 1)
19.A student solves 3x + 2y = 16 and 2x - y = 6 by substitution. They rearrange the second equation as y = 2x - 6 and substitute into the first. What equation do they get?
Medium
- A3x + 2(2x - 6) = 16
- B3x + 2(2x + 6) = 16
- C3x + 2(6 - 2x) = 16
- D3x + 2(2x - 6) = 6
20.Which of the following pairs (x, y) satisfy the equation 2x + 3y = 12? (select all that apply)
Medium
- A(0, 4)
- B(3, 2)
- C(6, 0)
- D(1, 3)
- E(2, 3)
21.Solve the simultaneous equations 3x + 2y = 16 and 2x - y = 6. What is the value of y?
Medium
- Ay = 2
- By = 4
- Cy = 3
- Dy = 1
22.Solve the simultaneous equations x + y = 5 and 2x - y = 1. What is the value of y?
Medium
- Ay = 3
- By = 2
- Cy = 1
- Dy = 4
23.Solve the simultaneous equations 3x + 2y = 16 and 2x - y = 6. What is the value of x - y?
Medium
- A2
- B1
- C3
- D0
24.Which method involves making the coefficients of one variable the same in both equations and then adding or subtracting the equations?
Medium
- AElimination
- BSubstitution
- CGraphical method
- DTrial and improvement
25.Which method involves rearranging one equation to make one variable the subject and then replacing that variable in the other equation?
Medium
- ASubstitution
- BElimination
- CGraphical method
- DTrial and improvement
26.On a graph, what does the point where two straight lines representing simultaneous equations cross represent?
Medium
- AThe solution to the simultaneous equations
- BThe y-intercept of both lines
- CThe gradient of both lines
- DThe origin
27.Solve the simultaneous equations x + y = 5 and 2x - y = 1. What is the value of x + y?
Medium
- A5
- B4
- C3
- D6
28.Solve the simultaneous equations x + y = 5 and 2x - y = 1. What is the value of x - y?
Medium
- A-1
- B1
- C0
- D-2
29.Which of these is a pair of simultaneous linear equations?
Easy
- Ax + y = 7 and 2x - y = 2
- Bx + y = 7 and y = x²
- Cx + y = 7
- Dx² + y = 7 and 2x - y = 2
30.Solve the simultaneous equations 4x + y = 14 and x + y = 5. What is the value of x?
Medium
- A3
- B2
- C4
- D1
31.Solve the simultaneous equations 5x + 2y = 24 and 3x − 2y = 8. What is the value of y?
Medium
- A1
- B2
- C-1
- D4
32.Solve the simultaneous equations 2x + 3y = 12 and x - y = 1. What is the value of x?
Medium
- A3
- B2
- C4
- D1
33.Solving a pair of simultaneous equations by elimination always involves adding the two equations together.
Easy
True or false?
34.Which of the following pairs (x, y) satisfy the equation 3x + 2y = 12? (select all that apply)
Medium
- A(2, 3)
- B(4, 0)
- C(0, 6)
- D(1, 4)
- E(3, 1.5)
35.Put the steps for solving the simultaneous equations 2x + 3y = 12 and x - y = 1 by substitution into the correct order.
Medium
- Rearrange x - y = 1 to make y the subject: y = x - 1
- Substitute y = x - 1 into 2x + 3y = 12 to get 2x + 3(x - 1) = 12
- Expand and solve for x: 5x - 3 = 12, so x = 3
- Substitute x = 3 into y = x - 1 to find y = 2
36.Which pair of simultaneous equations models this situation?
Medium
- A3p + 2c = 7 and 2p + 3c = 8
- B3p + 2c = 8 and 2p + 3c = 7
- C3p + 3c = 7 and 2p + 2c = 8
- D2p + 3c = 7 and 3p + 2c = 8
37.A rectangle has perimeter 26 cm. The length is 3 cm longer than the width. Let w be the width in cm and l be the length in cm. Which pair of simultaneous equations models this information?
Medium
- Al = w + 3 and 2l + 2w = 26
- Bl = w - 3 and 2l + 2w = 26
- Cl = w + 3 and l + w = 26
- Dl = 3w and 2l + 2w = 26
38.The sum of two numbers is 15 and their difference is 3. Let x be the larger number and y be the smaller number. Which pair of simultaneous equations models this information?
Medium
- Ax + y = 15 and x - y = 3
- Bx + y = 15 and y - x = 3
- Cx - y = 15 and x + y = 3
- Dx + y = 3 and x - y = 15
39.A rectangle has perimeter 26 cm and length 3 cm longer than its width. What is the area of the rectangle?
Hard
- A40 cm²
- B24 cm²
- C88 cm²
- D65 cm²
40.Two straight lines are y = 3x - 2 and y = x + 4. They cross at a point. What is the x-coordinate of this point?
Hard
- A3
- B1
- C2
- D6
41.Solve the simultaneous equations 2x + 3y = 6 and 4x + 9y = 15. Give the value of y.
Medium
42.Complete the sentence about solving simultaneous equations.
Medium
To solve x = 3 - (3/2)y and 4x + 9y = 15, you ____ the expression for x into the second equation.
43.What name is given to a collection of two or more linear equations involving the same variables?
Easy
44.Using your equations, find the cost of one pencil in pounds.
Medium
45.Solve the simultaneous equations 3x + 2y = 16 and 2x - y = 6. Give the value of y.
Medium
46.Solve the simultaneous equations x + y = 5 and 2x - y = 1. Give the value of y.
Medium
47.The lines y = 2x + 1 and y = -x + 4 intersect. Give the x-coordinate of the intersection point.
Hard
48.The lines y = 2x + 1 and y = -x + 4 intersect. Give the y-coordinate of the intersection point.
Hard
49.Solve the simultaneous equations 2x + y = 11 and x - y = 1. Give the value of x.
Medium
50.Complete the sentence about solving simultaneous equations by substitution.
Easy
To solve simultaneous equations by substitution, rearrange one equation to make one variable the ____, then substitute that expression into the other equation.
51.What name is given to the method where you make the coefficients of one variable the same in both equations and then add or subtract them?
Medium
52.Solve the equations to find the cost of one pen, in pounds.
Medium
53.Explain how you would solve the simultaneous equations 2x + y = 7 and x - y = 2 by elimination.
Medium
54.The sum of two numbers is 15 and their difference is 3. Explain how you would use simultaneous equations to find the two numbers.
Hard
55.A rectangle has a perimeter of 26 cm and a length 3 cm longer than its width. Explain how you would set up and solve simultaneous equations to find the dimensions.
Hard
56.Solve the simultaneous equations 2x + 3y = 6 and 4x + 9y = 15. Give the value of y.
Medium
- A1
- B0
- C2
- D3
57.Using your equations, find the cost of one pencil in pounds.
Medium
- A0 £
- B3 £
- C1 £
- D2 £
58.Solve the simultaneous equations 3x + 2y = 16 and 2x - y = 6. Give the value of y.
Medium
- A2
- B1
- C4
- D3
59.Solve the simultaneous equations x + y = 5 and 2x - y = 1. Give the value of y.
Medium
- A5
- B4
- C3
- D2
60.The lines y = 2x + 1 and y = -x + 4 intersect. Give the x-coordinate of the intersection point.
Hard
- A3
- B2
- C1
- D0
61.The lines y = 2x + 1 and y = -x + 4 intersect. Give the y-coordinate of the intersection point.
Hard
- A5
- B4
- C2
- D3
62.Solve the simultaneous equations 2x + y = 11 and x - y = 1. Give the value of x.
Medium
- A5
- B4
- C3
- D6
63.Solve the equations to find the cost of one pen, in pounds.
Medium
- A0 £
- B2 £
- C3 £
- D1 £

Ghi chú ôn tập

## Simultaneous Equations A **pair of simultaneous linear equations** is two equations involving the same two variables, e.g. x and y. A **solution** is a pair of values (x, y) that makes **both** equations true at the same time. ### Solving by substitution 1. Rearrange one equation to make one variable the subject. 2. Substitute that expression into the other equation. 3. Solve the resulting single-variable equation. 4. Substitute back to find the other variable. Example: \(2x + 3y = 6\) \(4x + 9y = 15\) From the first: \(x = 3 - \frac{3}{2}y\). Substitute into the second: \(4\left(3 - \frac{3}{2}y\right) + 9y = 15\). This gives \(y = 1\), then \(x = \frac{3}{2}\). ### Solving by elimination 1. Multiply one or both equations so the coefficients of one variable match. 2. Add or subtract the equations to remove that variable. 3. Solve for the remaining variable, then substitute back. ### Solving graphically Plot both straight lines. The solution is the **point where the two lines cross** — its coordinates are the values of x and y. ### Word problems - Define what each letter stands for. - Write two equations from the information. - Solve by substitution or elimination. - Check the answer fits the original problem (e.g. prices of two items). ### Key checks - Always substitute your values back into **both** original equations. - If the lines are parallel, there is no solution; if they are the same line, there are infinitely many solutions. ### Notation reminder A system can be written with a brace: \(\begin{cases} 2x + 3y = 6 \\ 4x + 9y = 15 \end{cases}\) A solution is an ordered pair \((x, y)\).

Simultaneous linear equations: A collection of two or more linear equations involving the same variables.
Solution: An assignment of values to the variables that makes all the equations simultaneously satisfied.
Substitution: Solving one equation for a variable and substituting that expression into the other equation.
Elimination: Adding or subtracting equations to remove one variable and solve for the other.
Graphical solution: The point where the two straight lines cross, giving the values of x and y.

Thẻ ghi nhớ

What is a pair of simultaneous linear equations?

Two linear equations that involve the same two variables, for example 2x + 3y = 6 and 4x + 9y = 15. A solution must make both equations true at the same time.

What does it mean to solve a pair of simultaneous equations?

Find the values of the variables that satisfy every equation in the system at the same time.

How can you solve simultaneous equations graphically?

Draw the graph of each equation as a straight line. The solution is the coordinates of the point where the two lines cross.

What does the point of intersection of two straight-line graphs represent?

It gives the x and y values that solve both equations simultaneously.

Describe the elimination method for solving simultaneous equations.

Add or subtract the two equations (after multiplying one or both if needed) to eliminate one variable, solve for the remaining variable, then substitute back to find the other.

Describe the substitution method for solving simultaneous equations.

Rearrange one equation to make one variable the subject, substitute that expression into the other equation, solve for the remaining variable, then substitute back to find the first variable.

In the substitution method, what do you do after finding the value of one variable?

Substitute it back into one of the original equations (or the rearranged one) to find the value of the other variable.

Solve by substitution: 2x + 3y = 6 and 4x + 9y = 15.

From the first equation, x = 3 − (3/2)y. Substituting into the second gives y = 1, then x = 3/2. So x = 3/2, y = 1.

Solve by elimination: x + y = 7 and x − y = 3.

Adding the equations gives 2x = 10, so x = 5. Substituting back gives y = 2. Solution: x = 5, y = 2.

When using elimination, why might you multiply one equation before adding or subtracting?

To make the coefficients of one variable the same (or opposites) so that adding or subtracting removes that variable.

How do you check that a solution to simultaneous equations is correct?

Substitute the values into both original equations. Both must be true.

What does it mean if two straight lines representing simultaneous equations are parallel?

They never meet, so there is no pair of values that satisfies both equations — the system has no solution.

What does it mean if two straight lines representing simultaneous equations are the same line?

Every point on the line satisfies both equations, so there are infinitely many solutions.

How do you form simultaneous equations from a word problem about the prices of two items?

Let one letter stand for the price of the first item and another for the second. Write one equation for each piece of information, then solve the pair.

Word problem: 3 pens and 2 pencils cost £7; 1 pen and 2 pencils cost £3. Form the equations.

Let p = cost of a pen and c = cost of a pencil. Then 3p + 2c = 7 and p + 2c = 3.

Solve: 3p + 2c = 7 and p + 2c = 3.

Subtract the second equation from the first: 2p = 4, so p = 2. Then 2 + 2c = 3 gives c = 0.5. So a pen costs £2 and a pencil costs 50p.

What is a linear equation?

An equation in which each variable appears only to the first power and is not multiplied by another variable, so its graph is a straight line.

In the system 2x + 3y = 6 and 4x + 9y = 15, what are the variables?

x and y.

Give the solution of the system 2x + 3y = 6 and 4x + 9y = 15.

x = 3/2 and y = 1.

Why is it useful to solve simultaneous equations by more than one method?

Different methods suit different systems; elimination is often quick when coefficients match, substitution is useful when one variable is already isolated, and a graph shows the solution visually.

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