Solving linear equations

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What a Linear Equation Is

A linear equation is an equation where the unknown appears only to the first power — no x², no √x, no 1/x.
In one variable it can be written as ax + b = 0, where a and b are numbers and a ≠ 0.
The unknown is the letter you are trying to find, usually x.
A solution is the value that makes the equation true when substituted for the unknown.
A one-variable linear equation with a ≠ 0 has exactly one solution.
For ax + b = 0 the solution is x = -\frac{b}{a}.

The Balance Method

Think of an equation as a balanced set of scales: whatever you do to one side, do to the other.
Adding or subtracting the same number on both sides keeps the equation balanced.
Multiplying or dividing both sides by the same non-zero number also keeps it balanced.
Your aim is to isolate the unknown — get x on its own on one side.
Check your answer by substituting it back into the original equation.

Inverse Operations

Inverse operations undo each other: addition undoes subtraction, multiplication undoes division.
To undo + 5, subtract 5 from both sides.
To undo − 3, add 3 to both sides.
To undo × 4, divide both sides by 4.
To undo ÷ 2, multiply both sides by 2.
Undo operations in reverse order: deal with + or − first, then × or ÷.

One-Step Equations

A one-step equation needs just one inverse operation to solve.
Example: x + 7 = 12 → subtract 7 from both sides → x = 5.
Example: x − 4 = 9 → add 4 to both sides → x = 13.
Example: 6x = 42 → divide both sides by 6 → x = 7.
Example: \frac{x}{5} = 3 → multiply both sides by 5 → x = 15.

Two-Step Equations

A two-step equation needs two inverse operations, done in the right order.
Example: 3x + 4 = 19 → subtract 4 → 3x = 15 → divide by 3 → x = 5.
Example: 2x − 5 = 11 → add 5 → 2x = 16 → divide by 2 → x = 8.
Example: \frac{x}{3} + 2 = 6 → subtract 2 → \frac{x}{3} = 4 → multiply by 3 → x = 12.
Always undo the + or − step before the × or ÷ step.

Equations with Brackets

Expand the bracket first, then solve as a two-step equation.
Example: 2(x + 3) = 14 → expand → 2x + 6 = 14 → subtract 6 → 2x = 8 → x = 4.
Example: 5(x − 2) = 20 → expand → 5x − 10 = 20 → add 10 → 5x = 30 → x = 6.
If a bracket is multiplied by a number, every term inside is multiplied by that number.
Alternatively, divide both sides by the number outside the bracket first, then solve.

Unknown on Both Sides

When the unknown appears on both sides, collect the x terms on one side.
Example: 5x = 2x + 9 → subtract 2x from both sides → 3x = 9 → x = 3.
Example: 4x − 3 = x + 6 → subtract x → 3x − 3 = 6 → add 3 → 3x = 9 → x = 3.
Move the smaller x term to avoid negative coefficients where possible.
After collecting, solve the remaining two-step equation as usual.

Equations with Fractions

To clear fractions, multiply every term by the denominator (or the lowest common multiple of the denominators).
Example: \frac{x}{2} + 1 = 4 → multiply all terms by 2 → x + 2 = 8 → x = 6.
Example: \frac{x+1}{3} = 2 → multiply both sides by 3 → x + 1 = 6 → x = 5.
Example: \frac{2x}{5} = 4 → multiply both sides by 5 → 2x = 20 → x = 10.
Multiplying by the denominator removes the fraction and leaves a simpler equation.

Forming and Solving Equations from Words

Form an equation by translating the words into algebra, using a letter for the unknown.
Look for key phrases: 'sum' means add, 'difference' means subtract, 'product' means multiply, 'quotient' means divide.
Example: 'I think of a number, multiply it by 3 and add 5. The result is 20.' → 3x + 5 = 20 → x = 5.
For geometry, use known facts: angles in a triangle sum to 180°, angles on a straight line sum to 180°.
Example: a triangle has angles x, 2x and 3x → x + 2x + 3x = 180 → 6x = 180 → x = 30.
After solving, interpret the answer in the context of the problem — check it makes sense.

Checking and Common Mistakes

Always substitute your solution back into the original equation to check both sides are equal.
A common mistake is forgetting to apply an operation to every term on both sides.
When expanding brackets, remember to multiply all terms inside, including the second one.
When collecting x terms, keep the equals sign balanced by doing the same to both sides.
If you get a negative coefficient, you can still solve by dividing by the negative number.

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1.Which of these is the correct first step to solve 5x + 3 = 18 using inverse operations?
Easy
- ASubtract 3 from both sides
- BDivide both sides by 5
- CAdd 3 to both sides
- DMultiply both sides by 5
2.Solve the equation 4x − 7 = 21.
Medium
- Ax = 7
- Bx = 3.5
- Cx = 14
- Dx = 28
3.Solve the equation 3(x + 4) = 27.
Medium
- Ax = 5
- Bx = 7
- Cx = 9
- Dx = 23
4.Solve the equation 5x + 2 = 3x + 10.
Medium
- Ax = 4
- Bx = 2
- Cx = 6
- Dx = 1.5
5.Solve the equation \frac{x}{3} + 2 = 7.
Medium
- Ax = 15
- Bx = 3
- Cx = 27
- Dx = 5
6.Solve the equation \frac{x}{4} - 1 = \frac{x}{2}.
Medium
- Ax = -4
- Bx = 4
- Cx = -2
- Dx = 2
7.The perimeter of a rectangle is 34 cm. The length is 3 cm more than the width. Form an equation and solve it to find the width.
Hard
- A7 cm
- B10 cm
- C17 cm
- D14 cm
8.Which of the following equations are linear equations in one variable? (Select all that apply.)
Medium
- A2x + 5 = 11
- Bx² − 4 = 0
- C3(x − 1) = 9
- D\frac{1}{x} = 2
- E5x = 0

Thêm câu hỏi

9.The equation 2x + 1 = 7 has exactly one solution.
Easy
True or false?
10.Which operation should be used first to solve the equation x + 8 = 15?
Easy
- AAdd 8 to both sides
- BSubtract 8 from both sides
- CMultiply both sides by 8
- DDivide both sides by 8
11.Solve the equation 6x = 42.
Easy
- Ax = 7
- Bx = 8
- Cx = 6
- Dx = 36
12.Solve the equation 9 − x = 4.
Medium
- Ax = 5
- Bx = −5
- Cx = 13
- Dx = −13
13.The equation 4x + 1 = 9 has exactly one solution.
Easy
True or false?
14.Match each equation with its solution.
Medium
- x + 7 = 10
- 3x = 18
- x − 4 = 5
- x/2 = 6
- x = 3
- x = 6
- x = 9
- x = 12
15.Put the steps for solving 4(x + 2) = 20 in the correct order.
Medium
- Divide both sides by 4
- Subtract 2 from both sides
- Expand the bracket: 4x + 8 = 20
- State the solution x = 3
16.Which equation represents the perimeter in terms of the width w?
Medium
- A2w + 2(w + 5) = 38
- Bw + (w + 5) = 38
- C2w + 5 = 38
- Dw(w + 5) = 38
17.Solve the equation x − 9 = 4.
Easy
- Ax = 13
- Bx = −5
- Cx = 5
- Dx = 36
18.Solve the equation 8x = 56.
Easy
- Ax = 7
- Bx = 8
- Cx = 48
- Dx = 64
19.Solve the equation 2x − 9 = 11.
Medium
- Ax = 10
- Bx = 1
- Cx = 20
- Dx = 2.5
20.Solve the equation 3(x − 2) = 18.
Medium
- Ax = 8
- Bx = 6
- Cx = 4
- Dx = 20
21.Solve the equation \frac{x}{5} + 3 = 9.
Medium
- Ax = 30
- Bx = 12
- Cx = 6
- Dx = 45
22.The equation x + 6 = 10 has exactly one solution.
Easy
True or false?
23.Put the steps for solving 5(x + 3) = 40 in the correct order.
Medium
- Divide both sides by 5 to get x + 3 = 8
- Subtract 3 from both sides to get x = 5
- Expand the bracket to get 5x + 15 = 40
- Check the solution by substituting x = 5 back into the equation
24.A rectangle has length 3 cm more than its width. Its perimeter is 26 cm. Which equation represents the perimeter in terms of the width w?
Medium
- A2w + 2(w + 3) = 26
- Bw + (w + 3) = 26
- C2w + (w + 3) = 26
- D2w + 2w + 3 = 26
25.A triangle has angles 2x°, 3x° and 4x°. What is the size of the largest angle?
Medium
- A80°
- B40°
- C100°
- D60°
26.A rectangle has width x cm and length (x + 4) cm. Its perimeter is 32 cm. Which equations are correct? (Select all that apply.)
Medium
- A2x + 2(x + 4) = 32
- B2(2x + 4) = 32
- Cx + (x + 4) = 32
- D4x + 8 = 32
- E2x + (x + 4) = 32
27.Which equation represents the statement: 'When 7 is added to three times a number n, the result is 25'?
Medium
- A3n + 7 = 25
- B3(n + 7) = 25
- C7n + 3 = 25
- D3n − 7 = 25
28.A taxi charges a fixed fee of £3 plus £2 per mile. A journey costs £17. Which equation gives the distance d in miles?
Medium
- A3 + 2d = 17
- B2 + 3d = 17
- C3d + 2d = 17
- D3 + d = 17
29.A shop sells pens for £2 each and pencils for £1 each. A student buys 5 items in total and spends £7. Which equation represents the number of pens p bought?
Medium
- A2p + (5 − p) = 7
- B2p + 5p = 7
- Cp + (5 − p) = 7
- D2p + 5 = 7
30.The sum of three consecutive integers is 42. Which equation represents the smallest integer n?
Medium
- An + (n + 1) + (n + 2) = 42
- Bn + (n + 1) + (n + 2) = 42n
- Cn + 2n + 3n = 42
- D3n = 42
31.A rectangle's length is twice its width. Its perimeter is 36 cm. Which equation represents the width w?
Medium
- A2w + 2(2w) = 36
- Bw + 2w = 36
- C2w + 2w = 36
- Dw + (w + 2) = 36
32.In the linear equation ax + b = 0 (a ≠ 0), what name is given to the letter a?
Easy
- AThe coefficient
- BThe variable
- CThe solution
- DThe constant term
33.A linear equation in one variable x can be written as ax + b = 0. Which condition must the coefficient a satisfy?
Easy
- Aa ≠ 0
- Ba = 0
- Ca > 0
- Da must be a whole number
34.A linear equation in one variable, ax + b = 0 with a ≠ 0, has exactly one solution.
Easy
True or false?
35.The equation 3x + 12 = 0 is written in the form ax + b = 0. What is its solution?
Medium
- Ax = 4
- Bx = −4
- Cx = −36
- Dx = 36
36.Which of the following statements about linear equations are correct? (Select all that apply.)
Medium
- AA linear equation in two variables has infinitely many solutions.
- BThe solutions of a linear equation in two variables form a straight line in the plane.
- CA linear equation in one variable always has two solutions.
- DThe graph of y = mx + c is a straight line.
- EA linear equation in two variables has exactly one solution.
37.Which of the following are linear equations in one variable? (Select all that apply.)
Medium
- A3x + 5 = 11
- Bx² + 2x = 8
- C2(x − 4) = 10
- D4/x = 2
- E7 = 2x − 1
38.The equation ax + by + c = 0 is a linear equation in two variables x and y. Which condition must the coefficients satisfy?
Medium
- Aa and b are not both 0
- Ba and b are both 0
- Cc must be 0
- Da must equal b
39.Match each equation with its correct solution.
Medium
- 7x = 42
- x + 11 = 20
- 5x - 4 = 31
- 3(x - 2) = 15
- x = 6
- x = 9
- x = 7
40.Put the steps for solving 6(x - 1) = 24 in the correct order.
Medium
- Divide both sides by 6 to get x - 1 = 4
- Add 1 to both sides to get x = 5
- Check: 6(5 - 1) = 6(4) = 24
- Write down the original equation 6(x - 1) = 24
41.Solve the equation 7x − 4 = 31.
Medium
42.Complete the sentence about solving equations.
Easy
In the equation 3x + 5 = 20, the first inverse operation is to ____ 5 from both sides.
43.Complete the sentence to describe the solution of a linear equation.
Medium
The ____ of an equation are the values that make the equality true.
44.Solve the equation 2x + 5 = 17.
Medium
45.What is the value of x in the equation 5x − 3 = 12?
Medium
46.Find the width of the rectangle in cm.
Medium
47.Solve the equation 4x + 5 = 29. Give your answer as a number.
Medium
48.Solve the equation 6x − 7 = 2x + 13. Give your answer as a number.
Medium
49.Complete the sentence describing how to solve 3x + 7 = 22.
Medium
First ____ 7 from both sides, then divide both sides by ____.
50.Using the equation 2w + 2(w + 3) = 26, find the width of the rectangle in cm.
Medium
51.Solve the equation 2x + 7 = 19. Give your answer as a number.
Easy
52.Solve the equation 5x - 3 = 2x + 12. Give your answer as a number.
Medium
53.Explain how you would solve the equation 2(x − 3) = 10 using inverse operations, and give the solution.
Medium
54.Explain how you would solve the equation 3(x − 1) = 12 and give the solution.
Medium
55.Explain how you would solve the equation 4(x + 2) = 28 using inverse operations, and give the solution.
Hard
56.Explain how you would solve the equation 2(x + 5) = 18 using inverse operations, and give the solution.
Hard
57.Explain how you would solve the equation 3(x - 2) = 15 and give the solution.
Hard
58.Solve the equation 7x − 4 = 31.
Medium
- A4
- B6
- C7
- D5
59.Solve the equation 2x + 5 = 17.
Medium
- A7
- B6
- C8
- D5
60.Find the width of the rectangle in cm.
Medium
- A6 cm
- B8 cm
- C9 cm
- D7 cm
61.Solve the equation 4x + 5 = 29. Give your answer as a number.
Medium
- A8
- B5
- C7
- D6
62.Solve the equation 6x − 7 = 2x + 13. Give your answer as a number.
Medium
- A5
- B6
- C7
- D4
63.Using the equation 2w + 2(w + 3) = 26, find the width of the rectangle in cm.
Medium
- A6 cm
- B4 cm
- C5 cm
- D7 cm
64.Solve the equation 2x + 7 = 19. Give your answer as a number.
Easy
- A6
- B8
- C5
- D7
65.Solve the equation 5x - 3 = 2x + 12. Give your answer as a number.
Medium
- A7
- B6
- C5
- D4

Ghi chú ôn tập

## Solving Linear Equations A **linear equation** has an unknown (usually x) with no powers higher than 1. In one variable it can be written as ax + b = 0, with a ≠ 0. The **solution** is the value of x that makes the equation true. ### The balance method An equation is like a balance: whatever you do to one side, do to the other. Undo operations using their **inverses**: - + and − are inverses - × and ÷ are inverses ### One-step equations Undo the single operation. - x + 7 = 12 → subtract 7 → x = 5 - 4x = 20 → divide by 4 → x = 5 ### Two-step equations Undo the + or − first, then the × or ÷. - 3x + 5 = 20 → subtract 5 → 3x = 15 → divide by 3 → x = 5 ### Brackets Expand first, then solve. - 2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4 ### Unknown on both sides Collect the x-terms on one side and numbers on the other. - 5x − 4 = 2x + 11 → subtract 2x → 3x − 4 = 11 → add 4 → 3x = 15 → x = 5 ### Fractions Multiply every term by the denominator (or the LCM of the denominators) to clear fractions. - \(\frac{x}{3} + 2 = 5\) → multiply by 3 → x + 6 = 15 → x = 9 ### Forming an equation 1. Name the unknown with a letter. 2. Translate the words or diagram into an equation. 3. Solve it. 4. Check the answer in the original context and give units. ### Checking Substitute your value back into the original equation. Both sides must be equal. ### Key fact For one variable, ax + b = 0 with a ≠ 0 has exactly one solution: \(x = -\frac{b}{a}\).

Linear equation: An equation that can be written in the form ax + b = 0 (one variable), where a ≠ 0; the unknown has no power higher than 1.
Unknown: The variable (often x) whose value you are finding.
Coefficient: The number multiplying the unknown, e.g. 3 in 3x.
Solution: A value that, when substituted for the unknown, makes the equation true.
Inverse operation: The operation that undoes another: + and −, × and ÷.
Balance method: Solving by doing the same operation to both sides so the equation stays balanced.

Thẻ ghi nhớ

What is a linear equation in one variable, written in general form?

ax + b = 0, where x is the unknown and a ≠ 0. The coefficients a and b are numbers that do not contain the variable.

What does it mean to 'solve' an equation?

To find the value(s) of the unknown that make the equality true when substituted in.

State the balance method for solving equations.

Whatever you do to one side of the equation, you must do to the other side, so the two sides stay equal.

What is the inverse operation of addition, and of multiplication?

The inverse of addition is subtraction; the inverse of multiplication is division.

Solve x + 7 = 12.

Subtract 7 from both sides: x = 5.

Solve x − 4 = 9.

Add 4 to both sides: x = 13.

Solve 5x = 35.

Divide both sides by 5: x = 7.

Solve x/3 = 8.

Multiply both sides by 3: x = 24.

Solve the two-step equation 3x + 4 = 19.

Subtract 4: 3x = 15. Divide by 3: x = 5.

Solve the two-step equation 2x − 5 = 11.

Add 5: 2x = 16. Divide by 2: x = 8.

What is the usual first step when solving an equation containing brackets, such as 4(x + 3) = 28?

Expand the brackets: 4x + 12 = 28. Then subtract 12 and divide by 4 to get x = 4.

Solve 5(x − 2) = 30.

Expand: 5x − 10 = 30. Add 10: 5x = 40. Divide by 5: x = 8.

How do you solve an equation with the unknown on both sides, such as 6x + 1 = 4x + 9?

Collect the x terms on one side by subtracting 4x: 2x + 1 = 9. Then subtract 1: 2x = 8, so x = 4.

Solve 7x − 3 = 3x + 13.

Subtract 3x: 4x − 3 = 13. Add 3: 4x = 16. Divide by 4: x = 4.

How do you solve an equation containing fractions, such as x/2 + x/3 = 5?

Multiply every term by the common denominator (6): 3x + 2x = 30, so 5x = 30 and x = 6.

Solve (x + 1)/4 = 3.

Multiply both sides by 4: x + 1 = 12. Subtract 1: x = 11.

What should you always do after solving an equation?

Substitute your answer back into the original equation to check both sides are equal.

Write an equation for: 'I think of a number, multiply it by 4, then add 5, and the result is 29.'

Let the number be n: 4n + 5 = 29. Solving gives 4n = 24, so n = 6.

A rectangle has length 3 cm more than its width w, and its perimeter is 26 cm. Form and solve an equation for w.

Perimeter: 2(w + w + 3) = 26, so 4w + 6 = 26. Then 4w = 20, so w = 5 cm (length 8 cm).

For the equation ax + b = 0 with a ≠ 0, what is the solution for x?

x = -\frac{b}{a}

In the general linear equation ax + b = 0, what must be true about a?

a ≠ 0. If a were 0 the equation would not be linear and would have no single solution.

How many solutions does a linear equation in one variable have (when a ≠ 0)?

Exactly one solution.

How many solutions does a linear equation in two variables, such as ax + by + c = 0, have?

Infinitely many; each solution (x, y) is a point on a straight line.

Why is an equation like ax + by + c = 0 called 'linear'?

Because its solutions form a straight line in the plane.

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