Linear Equations

Aprenda jogando

Responda a estas perguntas para ganhar energia, depois pesque e explore. Sem precisar de conta.

Para professores: slides de aula, resumos de revisão, diagramas prontos para usar de Linear Equations (Maths [CIE], Core) — use na sua aula ou rode o tópico como atividade interativa de turma que seus alunos jogam como um jogo ao vivo.

Notas de aula

Solving Linear Equations

A linear equation can be written as ax + b = c, where the highest power of x is 1.
To solve, use inverse operations: add/subtract to undo addition/subtraction; multiply/divide to undo multiplication/division.
Any operation performed on one side must be done to the other side.
For 2x + 1 = 9, subtract 1 from both sides to get 2x = 8, then divide by 2 to get x = 4.
For equations with negative terms, e.g., 2 - 3x = 10, add 3x to both sides to make the x term positive, then solve.
Always check your answer by substituting it back into the original equation.

Equations with Brackets & Fractions

If an equation contains brackets, expand the brackets first: e.g., 2(x - 3) = 10 becomes 2x - 6 = 10.
Alternatively, you can divide both sides by the number outside the bracket: 2(x - 3) = 10 becomes x - 3 = 5.
If an equation contains fractions, multiply both sides by the lowest common denominator (LCD).
For \frac{x}{5} + 4 = \frac{9}{2}, the LCD is 10. Multiply all terms by 10: 2x + 40 = 45, then solve.
If the unknown is in the denominator, e.g., \frac{4}{x-2} = 3, multiply both sides by the denominator: 4 = 3(x-2), then solve.

Equations with Unknowns on Both Sides

Collect the x terms on one side by adding or subtracting the smaller x term from both sides.
For 4x - 7 = 11 + x, subtract x from both sides to get 3x - 7 = 11, then solve.
For 4 - 5x = 6x - 29, add 5x to both sides to get 4 = 11x - 29, then solve.
After solving, reflect the equation if necessary to present the answer as x = ....

Forming Equations from Words

Use a variable (e.g., x) to represent the unknown quantity.
Translate phrases into expressions: '2 less than' → x - 2; 'double' → 2x; 'half of' → \frac{x}{2}.
Use brackets to maintain order: 'add 1 then multiply by 3' → 3(x+1).
An equation is a statement with an equals sign; identify where 'is equal to' fits.
Always answer in context: after solving, state the value of the required quantity.

Forming Equations from Shapes

Use properties of shapes (perimeter, area, angles) to set up equations.
For a rectangle: perimeter = 2(\text{length} + \text{width}), area = \text{length} \times \text{width}.
For triangles: sum of interior angles = 180°. For polygons: sum = 180(n-2).
Substitute algebraic expressions into formulas, using brackets where needed.
Read the question carefully to determine whether to find an angle, length, area, etc.

Rectangle

Slides

Step through every slide for this topic — plus flashcards and revision notes — with a free account.

CADASTRE-SE — GRÁTIS ENTRAR

Questões de prática

Prévia grátis — 8 de 54 perguntas. Cadastre-se para ver todas.

1.Solve the equation 12x - 7 = 23.
Easy
- Ax = 2.5
- Bx = 0.5
- Cx = 30
- Dx = 1.33
2.Solve x + 7 = 15.
Easy
- Ax = 8
- Bx = 22
- Cx = 105
- Dx = 7
3.Solve 5(3x + 8) = 10.
Medium
- Ax = -2
- Bx = 2
- Cx = -0.5
- Dx = 0.5
4.Solve 8(w + 11) = 120.
Easy
- Aw = 4
- Bw = 15
- Cw = 26
- Dw = 1
5.Solve (x - 2)/3 = 3.
Medium
- Ax = 11
- Bx = 7
- Cx = 5
- Dx = 1
6.Solve 5x + 18 = 8.
Easy
- Ax = -2
- Bx = 5.2
- Cx = -10
- Dx = 2
7.Solve 12x - 3 = 4x + 21.
Medium
- Ax = 3
- Bx = 2.25
- Cx = 1.5
- Dx = 24
8.Solve 6 - 2x = 3x.
Medium
- Ax = 1.2
- Bx = 6
- Cx = -6
- Dx = 0.6

Mais perguntas

9.Solve 5(2x + 4) = 85.
Medium
- Ax = 6.5
- Bx = 8.5
- Cx = 7.5
- Dx = 9
10.A rectangle has length (3x + 1) cm and width (2x - 5) cm. Its perimeter is 22 cm. Find x.
Medium
- Ax = 3
- Bx = 4
- Cx = 2
- Dx = 5
11.Rovers scored x goals. United scored 8 more than Rovers. City scored 3 less than twice Rovers. Total goals = 117. Find x.
Hard
- Ax = 28
- Bx = 25
- Cx = 30
- Dx = 32
12.The total cost for Speedy Coaches is 15x + 625. Sporty Coaches charges 19x + 117. Find x when costs are equal.
Medium
- Ax = 127
- Bx = 100
- Cx = 150
- Dx = 80
13.Cara has n pencils. Alice has twice as many as Cara. Leon has three more than Alice. Total pencils = 58. Find n.
Medium
- An = 11
- Bn = 10
- Cn = 12
- Dn = 13
14.Esme buys 8 magazines at $2.45 each and y cards at $3.15 each. Total $60.55. How many cards?
Hard
- Ay = 13
- By = 12
- Cy = 14
- Dy = 11
15.Neelum hires a machine: $25 first day, $9 each extra day. Total paid $88. Total days hired?
Medium
- A8 days
- B7 days
- C9 days
- D10 days
16.Solve 8T = 12 + 5T.
Easy
- AT = 4
- BT = 2.4
- CT = 1.5
- DT = 6
17.Solve the equation 7x - 5 = 16.
Easy
- Ax = 3
- Bx = 2
- Cx = 1
- Dx = 4
18.Solve the equation x/3 = 18.
Easy
- Ax = 54
- Bx = 6
- Cx = 15
- Dx = 21
19.Solve 5x = -30.
Easy
- Ax = -6
- Bx = 6
- Cx = -25
- Dx = 35
20.Solve 4x - 2 = 28.
Easy
- Ax = 7.5
- Bx = 6.5
- Cx = 30
- Dx = 26
21.Solve 3(2x + 7) = 12.
Medium
- Ax = -1.5
- Bx = 1.5
- Cx = -3
- Dx = 3
22.Solve 10/x = 5.
Easy
- Ax = 2
- Bx = 0.5
- Cx = 50
- Dx = 15
23.Solve 7x - 3 = 2x + 11.
Medium
- Ax = 2.8
- Bx = 14
- Cx = 1.6
- Dx = 8
24.Solve 3(2x - 1) = 27.
Medium
- Ax = 5
- Bx = 4
- Cx = 14
- Dx = 13
25.Solve x/4 = 20.
Easy
- Ax = 80
- Bx = 5
- Cx = 16
- Dx = 24
26.Solve 3x - 5 = 16.
Easy
- Ax = 7
- Bx = 21
- Cx = 11
- Dx = 5.67
27.Solve 5(2x + 1) = 27.
Medium
- Ax = 2.2
- Bx = 2.6
- Cx = 3.2
- Dx = 2.8
28.Solve 5(x - 1) = 3x + 7.
Medium
- Ax = 6
- Bx = 3
- Cx = 1
- Dx = 4
29.Solve 2x + 3 = 11.
Easy
- Ax = 4
- Bx = 7
- Cx = 5.5
- Dx = 8
30.Solve 3x - 7 = 8.
Easy
- Ax = 5
- Bx = 15
- Cx = 1
- Dx = 3
31.Solve 4(x + 2) = 20.
Easy
- Ax = 3
- Bx = 4.5
- Cx = 18
- Dx = 5
32.Solve 2(3x - 4) = 5x + 3.
Medium
- Ax = 11
- Bx = 7
- Cx = 5
- Dx = 1
33.A rectangle has length (3x+2) cm and width (2x-1) cm. Its perimeter is 42 cm. Find the value of x.
Hard
- A4
- B5
- C6
- D7
34.Solve the equation 3(2x-5) = 4x+7.
Medium
- A11
- B10
- C9
- D8
35.Solve the equation 5x/4 = 1/2.
Medium
- A2/5
- B5/2
- C2
- D5
36.What is the inverse operation of addition?
Easy
- ASubtraction
- BMultiplication
- CDivision
- DAddition
37.In the equation 3x + 5 = 11, the first step to solve for x is to subtract 5 from both sides.
Easy
True or false?
38.Solve 4x = 20.
Easy
- Ax = 5
- Bx = 16
- Cx = 80
- Dx = 24
39.Solve 5x - 3 = 2x + 11.
Medium
- A4.67
- B4
- C14/3
- D8/3
40.Put the following steps in the correct order to solve 2x + 3 = 11.
Easy
- Subtract 3 from both sides
- Divide both sides by 2
- Simplify to 2x = 8
- Simplify to x = 4
41.Which of the following are correct steps to solve 3(x - 2) = 12? (Select all that apply)
Medium
- ADivide both sides by 3 first to get x - 2 = 4
- BExpand brackets to get 3x - 6 = 12
- CAdd 2 to both sides first to get 3x = 14
- DSubtract 6 from both sides first
42.Match each operation on the left with its inverse operation on the right.
Easy
- Addition
- Subtraction
- Multiplication
- Division
- Subtraction
- Addition
- Division
- Multiplication
43.Put the steps in the correct order to solve 3x + 5 = 20.
Easy
- Subtract 5 from both sides
- Divide both sides by 3
- Simplify: 3x = 15
- Simplify: x = 5
44.Which of the following are correct steps to solve 2(x - 3) = 8? (Select all that apply)
Medium
- AExpand brackets: 2x - 6 = 8
- BDivide both sides by 2 first: x - 3 = 4
- CAdd 6 to both sides: 2x = 14
- DSubtract 6 from both sides: 2x = 2
45.To solve 5x - 3 = 2x + 9, the first step is to subtract 2x from both sides.
Medium
True or false?
46.Complete the sentence: The solution to the equation x + 7 = 15 is x = ___.
Easy
The solution to the equation x + 7 = 15 is x = ____.
47.Solve the equation 3x - 5 = 16.
Easy
48.Form an equation from the following: A number is multiplied by 3 and then 5 is added to give 20. Find the number.
Medium
49.Solve the equation (x - 2)/3 = 3.
Medium
50.Solve the equation 3x/5 = 9.
Easy
51.Form an equation from the following and solve: A number is multiplied by 4, then 7 is subtracted to give 21. Find the number.
Medium
Equation: ____ = 21. Number = ____.
52.A rectangle has length (3x + 1) cm and width (2x - 5) cm. Its perimeter is 22 cm. Find the value of x and the area of the rectangle.
Hard
53.A triangle has angles (2x)°, (3x)°, and (x + 20)°. Write an equation and solve for x. Hence find the size of the largest angle.
Hard
54.Solve the equation 3x/5 = 9.
Easy
- A17
- B14
- C16
- D15

Resumos de revisão

## Linear Equations - Quick Revision ### Solving Linear Equations - A linear equation has the form $ax + b = c$ with $x$ to power 1. - **Inverse operations**: add/subtract, multiply/divide. Do the same to both sides. - **Order**: usually undo addition/subtraction first, then multiplication/division. - Example: $2x + 1 = 9$ → subtract 1: $2x = 8$ → divide by 2: $x = 4$. - For negative terms, add the negative term to both sides. E.g., $2 - 3x = 10$ → add $3x$: $2 = 10 + 3x$ → rewrite as $3x + 10 = 2$ → solve. ### Equations with Brackets & Fractions - **Brackets**: expand first, then solve. E.g., $2(x - 3) = 10$ → $2x - 6 = 10$ → $2x = 16$ → $x = 8$. - **Fractions**: multiply both sides by the lowest common denominator (LCD). E.g., $\frac{x}{5} + 4 = \frac{9}{2}$ → LCD = 10 → $2x + 40 = 45$ → $2x = 5$ → $x = \frac{5}{2}$. - If unknown is in denominator, multiply both sides by that denominator. E.g., $\frac{4}{x-2} = 3$ → $4 = 3(x-2)$ → solve. ### Equations with Unknowns on Both Sides - Collect $x$ terms on one side by adding/subtracting the smaller $x$ term. E.g., $4x - 7 = 11 + x$ → subtract $x$: $3x - 7 = 11$ → add 7: $3x = 18$ → $x = 6$. - For $4 - 5x = 6x - 29$ → add $5x$: $4 = 11x - 29$ → add 29: $33 = 11x$ → $x = 3$. ### Forming Equations from Words - Translate phrases: "2 less than something" → $x - 2$; "double" → $2x$; "sum" → add. - Choose a variable for the unknown, then write an equation with an equals sign. - Example: "Rovers scored $x$ goals, United scored 8 more, City scored 3 less than twice Rovers, total 117" → $x + (x+8) + (2x-3) = 117$ → solve: $4x + 5 = 117$ → $4x = 112$ → $x = 28$. ### Forming Equations from Shapes - Use perimeter, area, angle sums, etc. E.g., rectangle perimeter: $2(\text{length}) + 2(\text{width})$. - Example: rectangle length $3x+1$, width $2x-5$, perimeter 22 → $2(3x+1) + 2(2x-5) = 22$ → $10x - 8 = 22$ → $x = 3$. - Always substitute back to check. **Key Tip**: Substitute your answer into the original equation to verify.

Linear equation: An equation that can be written in the form ax + b = c, where a, b, c are numbers and x is the variable with the greatest power of 1.
Inverse operations: Operations that undo each other: addition and subtraction are inverses; multiplication and division are inverses.
Lowest common denominator (LCD): The smallest multiple that two or more denominators share; used to eliminate fractions by multiplying both sides of the equation by the LCD.
Expand brackets: To multiply each term inside the brackets by the term outside, e.g., 2(x - 3) = 2x - 6.
Form an equation: To translate a word problem or geometric situation into a mathematical statement with an equals sign that can be solved.

Flashcards

What is a linear equation?

A linear equation can be written in the form ax + b = c, where a, b, c are numbers and x is the variable. The greatest power of x is 1.

How do you solve 2x + 1 = 9?

Subtract 1 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4.

What are inverse operations in solving equations?

Addition and subtraction are inverse operations; multiplication and division are inverse operations.

Solve 4x + 8 = 12.

Subtract 8 from both sides: 4x = 4. Then divide by 4: x = 1.

Solve 2 - 3x = 10.

Add 3x to both sides: 2 = 10 + 3x. Subtract 10: -8 = 3x. Divide by 3: x = -8/3.

Solve 5x - 8 = 22.

Add 8 to both sides: 5x = 30. Divide by 5: x = 6.

How do you solve a linear equation with brackets?

Expand the brackets first. For example, 2(x - 3) = 10 becomes 2x - 6 = 10, then solve as usual.

Solve 2(x - 3) = 10.

Expand: 2x - 6 = 10. Add 6: 2x = 16. Divide by 2: x = 8.

How do you solve a linear equation with fractions?

Multiply both sides by the lowest common denominator to eliminate fractions. Then solve as usual.

Solve x/5 + 4 = 9/2.

Multiply by 10: 2x + 40 = 45. Subtract 40: 2x = 5. Divide by 2: x = 5/2.

Solve 4/(x - 2) = 3.

Multiply both sides by (x - 2): 4 = 3(x - 2). Expand: 4 = 3x - 6. Add 6: 10 = 3x. Divide: x = 10/3.

How do you solve an equation with unknowns on both sides?

Collect x terms on one side by adding or subtracting the smaller x term from both sides.

Solve 4x - 7 = 11 + x.

Subtract x from both sides: 3x - 7 = 11. Add 7: 3x = 18. Divide by 3: x = 6.

Solve 4 - 5x = 6x - 29.

Add 5x to both sides: 4 = 11x - 29. Add 29: 33 = 11x. Divide by 11: x = 3.

How do you form an expression from '2 less than something'?

x - 2, where x is the unknown.

How do you form an expression from 'double something'?

2x, where x is the unknown.

How do you form an expression from 'half of something'?

x/2 or (1/2)x, where x is the unknown.

How do you form an expression from 'something add 1, then multiplied by 3'?

(x + 1) × 3, which simplifies to 3(x + 1).

The number of yellow flowers is three times the number of red flowers. Purple flowers are 5 more than yellow. The difference between purple and red is 29. Find the number of yellow flowers.

Let red = x. Then yellow = 3x, purple = 3x + 5. Equation: (3x + 5) - x = 29 → 2x + 5 = 29 → 2x = 24 → x = 12. Yellow = 3 × 12 = 36.

A rectangle has length 3x + 1 cm and width 2x - 5 cm. Its perimeter is 22 cm. Find x.

Perimeter = 2(3x+1) + 2(2x-5) = 6x+2+4x-10 = 10x-8. Set equal to 22: 10x-8=22 → 10x=30 → x=3.

For the rectangle with x=3, length 3x+1 and width 2x-5, find the area.

Length = 3(3)+1 = 10 cm, width = 2(3)-5 = 1 cm. Area = 10 × 1 = 10 cm².

Solve 5(3x + 8) = 10.

Expand: 15x + 40 = 10. Subtract 40: 15x = -30. Divide by 15: x = -2.

Solve 8(w + 11) = 120.

Divide both sides by 8: w + 11 = 15. Subtract 11: w = 4.

Solve (x - 2)/3 = 3.

Multiply by 3: x - 2 = 9. Add 2: x = 11.

Solve 7x - 5 = 16.

Add 5: 7x = 21. Divide by 7: x = 3.

Unlock all 54 questions, flashcards & more

Crie uma conta grátis para ver todas as perguntas, os slides, flashcards e resumos de revisão deste tópico.

CADASTRE-SE — GRÁTIS ENTRAR

Provas anteriores

A prática com provas anteriores deste tópico chega em breve.

Em breve