Linear Equations

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Notes

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 $= 180^{\circ}$ . 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.

Solving a Linear Equation Step by Step

Expanding Brackets

Collecting Like Terms

Forming an Equation from a Shape

Practice questions

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1.Solve the equation $12x - 7 = 23$ .
Easy
- A $x = 2.5$
- B $x = 0.5$
- C $x = 30$
- D $x = 1.33$
2.Solve $x + 7 = 15$ .
Easy
- A $x = 8$
- B $x = 22$
- C $x = 105$
- D $x = 7$
3.Solve $5(3x + 8) = 10$ .
Easy
- A $x = -2$
- B $x = 2$
- C $x = -0.5$
- D $x = 0.5$
4.Solve $8(w + 11) = 120$ .
Easy
- A $w = 4$
- B $w = 15$
- C $w = 26$
- D $w = 1$
5.Solve $(x - 2)/3 = 3$ .
Easy
- A $x = 11$
- B $x = 7$
- C $x = 5$
- D $x = 1$
6.Solve $5x + 18 = 8$ .
Medium
- A $x = -2$
- B $x = 5.2$
- C $x = -10$
- D $x = 2$
7.Solve $12x - 3 = 4x + 21$ .
Medium
- A $x = 3$
- B $x = 2.25$
- C $x = 1.5$
- D $x = 24$
8.Solve $6 - 2x = 3x$ .
Medium
- A $x = 1.2$
- B $x = 6$
- C $x = -6$
- D $x = 0.6$

Summary

## 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.

Key terms

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.

Slides

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 highest power of x is $1 (no x^{2} terms)$ .
Examples: $2x + 3 = 5, 3x + 4 = 1, x - 5 = -3$ .

Which of these is a linear equation?

$x^{2} + 2 = 5$
$3x - 1 = 7$
$\frac{1}{x} = 2$
$\sqrt{x} = 4$

Solving Linear Equations

Use inverse operations to get x alone.
Addition and subtraction are inverse operations.
Multiplication and division are inverse operations.
Whatever you do to one side, do to the other.

To solve $2x + 1 = 9$ , first subtract ____ from both sides, then divide $by ____$ .

Order of Operations

It's easier to undo addition/subtraction before multiplication/division.
Example: $4x + 8 = 12$ → subtract 8 first, then divide by 4.
If you divide first, you must divide every term by 4.

Solve $5x - 8 = 22$

$x = 6$
$x = 5$
$x = 7$
$x = 4$

Equations with Brackets

Expand brackets first: $2(x - 3) = 10$ becomes $2x - 6 = 10$ .
Alternatively, divide both sides by the number outside the bracket first.
Expanding first always works.

To solve $5(2x + 4) = 85$ , first expand to $get ____ = 85$ , then subtract 20 and divide by 10.

Equations with Fractions

Multiply both sides by the lowest common denominator (LCD).
Example: $\frac{x}{5} + 4 = \frac{9}{2}$ → multiply by $10: 2x + 40 = 45$ .
Then solve as usual.

Solve $\frac{x}{3} = 18$

$x = 6$
$x = 54$
$x = 15$
$x = 21$

Unknowns on Both Sides

Collect x terms on one side by adding or subtracting the smaller x term.
Example: $4x - 7 = 11 + x$ → subtract x from both sides: $3x - 7 = 11$ .
Then solve: $3x = 18, x = 6$ .

Which steps are correct for solving $4 - 5x = 6x - 29$ ?

Add 5x to both sides: $4 = 11x - 29$
Add 29 to both sides: $33 = 11x$
Divide by $11: x = 3$
Subtract 4 from both sides first

Forming Equations from Words

Use x for the unknown.
Translate phrases: 'double' → 2x, '3 more than' → $x + 3, 'half'$ → x/2.
Look for 'is equal to' to place the equals sign.

Rovers scored x goals. United scored 8 more than Rovers, so United scored ____. City scored 3 less than twice Rovers, so City scored ____. Total is 117, so equation: $x + (x+8) + (2x-3) = 117$ .

If Cara has n pencils, Alice has twice as many, Leon has three more than Alice, and total is 58, what is the equation?

$n + 2n + (2n+3) = 58$
$n + 2n + 3 = 58$
$n + 2n + 2n+3 = 58$
$n + 2n + 3n = 58$

Forming Equations from Shapes

Use properties of shapes (perimeter, area, angles).
Substitute algebraic expressions into formulas.
Example: rectangle length 3x+1, width 2x-5, perimeter 22 → $2(3x+1)+2(2x-5)=22$ .

A rectangle has length 3x+1 and width 2x-5. Its perimeter is 22. What is the value of x?

$x = 3$
$x = 4$
$x = 5$
$x = 2$

Which of these are correct steps for solving $5(x-1)=3x+7$ ?

Expand: $5x-5=3x+7$
Subtract $3x: 2x-5=7$
Add $5: 2x=12$
Divide by $2: x=6$

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 $3: x = -\frac{8}{3}$ .

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 $\frac{x}{5} + 4 = \frac{9}{2}$ .

Multiply by $10: 2x + 40 = 45$ . Subtract $40: 2x = 5$ . Divide by $2: x = \frac{5}{2}$ .

Solve $4/(x - 2) = 3$ .

Multiply both sides by $(x - 2): 4 = 3(x - 2)$ . Expand: $4 = 3x - 6$ . Add $6: 10 = 3x$ . Divide: $x = \frac{10}{3}$ .

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 $(\frac{1}{2})x$ , where x is the unknown.

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

$(x + 1) \times 3$ , 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 $= 3 \times 12 = 36$ .

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 $= 10 \times 1 = 10 cm^{2}$ .

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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