Linear Equations

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レッスンノート

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

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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

もっと問題

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

復習ノート

## 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.

フラッシュカード

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.

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