Linear equations

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レッスンノートスライド練習問題

レッスンノート

What a linear equation is

A linear equation contains an unknown (such as x) to the power 1 only, with no x² or \frac{1}{x} terms.
To solve it means to find the value of the unknown that makes both sides equal.
Keep the equation balanced: whatever you do to one side you must do to the other.

Solving one- and two-step equations

Use inverse operations to undo what has been done to x: the inverse of + is −, of × is ÷.
Example: 3x + 5 = 20. Subtract 5: 3x = 15. Divide by 3: x = 5.
Check by substituting back into the original equation.

Unknowns on both sides and brackets

Expand any brackets first: 2(x + 3) = 2x + 6.
Collect the x terms on one side, then the numbers on the other. Move the smaller x term to avoid negative coefficients.
Example: 5x - 4 = 2x + 11 gives 3x = 15, so x = 5.

Equations with fractions

Multiply every term by the common denominator to clear the fractions.
Example: \frac{x}{4} + 2 = 5 gives \frac{x}{4} = 3, so x = 12.
For \frac{x+1}{3} = \frac{x-2}{2}, multiply both sides by 6: 2(x+1) = 3(x-2).

Common errors

Changing a sign wrongly when moving a term across the equals sign.
Dividing only one term instead of the whole side.
Forgetting to multiply every term when clearing fractions, or not expanding a bracket fully (-2(x-3) = -2x + 6, not -2x - 6).

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練習問題

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1.What does it mean to solve an equation?
Easy
- ARewrite it in a different order
- BFind the value of the unknown that makes both sides equal
- CExpand all the brackets
- DDraw its graph
2.Solve x + 9 = 15.
Easy
- Ax = 6
- Bx = 24
- Cx = -6
- Dx = 135
3.Solve 4x = 28.
Easy
- Ax = 24
- Bx = 7
- Cx = 32
- Dx = 112
4.Solve x - 8 = 3.
Easy
- Ax = -5
- Bx = 5
- Cx = 11
- Dx = 24
5.Solve 2x + 7 = 19.
Easy
- Ax = 6
- Bx = 13
- Cx = 5
- Dx = 26
6.Solve 5x - 2 = 18.
Easy
- Ax = 3.2
- Bx = 4
- Cx = 16
- Dx = 90
7.Solve 3x + 4 = 1.
Easy
- Ax = 1
- Bx = -1
- Cx = \frac{5}{3}
- Dx = -\frac{5}{3}
8.Solve \frac{x}{3} + 1 = 6.
Easy
- Ax = 7
- Bx = 15
- Cx = 21
- Dx = 5

もっと問題

9.Solve 6 - x = 2.
Easy
- Ax = -4
- Bx = 8
- Cx = 4
- Dx = -8
10.Solve 6x - 5 = 2x + 11.
Medium
- Ax = 1.5
- Bx = 4
- Cx = 6
- Dx = 16
11.Solve 3(x + 2) = 21.
Medium
- Ax = 5
- Bx = 7
- Cx = 3
- Dx = 19
12.Solve 2(3x - 1) = 22.
Medium
- Ax = 4
- Bx = 3.67
- Cx = 12
- Dx = 2
13.Solve 8 - 3x = 2.
Medium
- Ax = -2
- Bx = 2
- Cx = \frac{10}{3}
- Dx = 6
14.Solve 4x + 3 = 7x - 9.
Medium
- Ax = 4
- Bx = -4
- Cx = 2
- Dx = 12
15.Solve 5(x - 2) = 3(x + 4).
Medium
- Ax = 11
- Bx = 1
- Cx = 6
- Dx = 7
16.Solve \frac{x + 1}{3} = 4.
Medium
- Ax = 11
- Bx = 12
- Cx = 13
- Dx = 3
17.Solve \frac{2x}{5} = 6.
Medium
- Ax = 2.4
- Bx = 12
- Cx = 15
- Dx = 30
18.Solve \frac{x}{2} + \frac{x}{3} = 10.
Medium
- Ax = 12
- Bx = 6
- Cx = 50
- Dx = 5
19.Solve \frac{x + 1}{3} = \frac{x - 2}{2}.
Medium
- Ax = 8
- Bx = 4
- Cx = 1
- Dx = 7
20.Expanding -2(x - 3) gives:
Medium
- A-2x - 6
- B-2x + 6
- C2x - 6
- D-2x + 3
21.A number is multiplied by 3 and then 4 is added to give 22. What is the number?
Easy
- A6
- B8
- C26
- D5.5
22.The perimeter of a rectangle is 30 cm. Its length is x + 5 and its width is x. What is x?
Medium
- A5
- B10
- C7.5
- D2.5
23.Three consecutive integers add up to 72. What is the smallest?
Medium
- A23
- B24
- C25
- D22
24.Why must you do the same operation to both sides of an equation?
Easy
- ATo make the numbers smaller
- BTo keep the equation balanced
- CTo remove the unknown
- DSo that the answer is positive
25.In 5x - 4 = 2x + 11, what is the best first step?
Medium
- ASubtract 2x from both sides
- BDivide by 5
- CAdd x to both sides
- DSubtract 11 from the left only
26.The equation x² = 9 is a linear equation.
Easy
True or false?
27.The solution of x + 4 = 10 is x = 6.
Easy
True or false?
28.You can check a solution by substituting it back into the original equation.
Easy
True or false?
29.If 3x = 12, then x = 36.
Easy
True or false?
30.2(x + 3) = 2x + 3.
Medium
True or false?
31.Which of these equations have the solution x = 3?
Medium
- A2x + 1 = 7
- Bx - 1 = 4
- C5x = 15
- Dx + 6 = 8
- E4x - 2 = 10
32.Which two inverse operations, in this order, solve x/4 + 2 = 5 by undoing the operations applied to x? Select both.
Medium
- ASubtract 2 from both sides
- BMultiply both sides by 4
- CAdd 2 to both sides
- DDivide both sides by 4
33.Solve 4(x - 1) = 2x + 8, showing each step.
Medium

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