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# Algebra Solving Linear Equations

2x + 3 = 7

## Distinguish between an algebraic expression and an algebraic equation

##### what you need to know

An algebraic expression is a combination of variables, constants, and operations, without an equal sign, representing a value. An algebraic equation, on the other hand, includes an equal sign and is used to find the value of the variable(s) that satisfy the equation.

Algebraic Expression:

$3x^2 + 2y - 5$

This is an algebraic expression because it consists of terms involving variables $$x$$ and $$y$$, along with constants and operations. It is not stating that the expression is equal to anything; it is just a representation of a mathematical expression.

Algebraic Equation

$4x + 7 = 15$

This is an algebraic equation since it has an equal sign and the aim is to find the value of the variable $$x$$ that satisfies the equation.

Worked Example

Given the equation:

$2x + 5 = 11$

Let's solve this equation to find the value of $$x$$.

Subtract 5 from both sides:

$2x + 5 - 5 = 11 - 5$

$2x = 6$

Divide by 2:

$\frac{2x}{2} = \frac{6}{2}$

$x = 3$

Therefore, the solution to the equation $$2x + 5 = 11$$ is $$x = 3$$.

## Solving linear equations with one unknown, including those involving brackets and fractions

##### what you need to know
• Basic Linear Equations: These are straightforward equations like $$2x + 3 = 7$$, where you isolate $$x$$ by performing inverse operations (subtracting, dividing, etc.) on both sides of the equation until $$x$$ is alone on one side.
• Equations with Brackets: Equations such as $$3(x + 2) - 2 = 10$$ involve distributing or expanding the terms within the brackets first, before simplifying the equation further.
• Equations with Fractions: Equations like $$\frac{2x}{3} + \frac{x - 1}{2} = 4$$ require clearing fractions by multiplying through by the least common denominator of all fractions involved.
• Combination of Brackets and Fractions: Equations such as $$2(3x - \frac{1}{2}) = 5$$ require both distribution and handling of fractions in the process of solving for $$x$$.

Note: Solving linear equations with one unknown, whether they involve brackets, fractions, or both, requires systematic application of algebraic rules and operations to isolate the variable and determine its value.

## Practical problems involving simple linear equations

Practical problems involving simple linear equations often require translating a real-world scenario into an algebraic equation that can be solved to find the unknown value. Here’s how we approach and differentiate these problems:

##### what you need to know
1. Basic Linear Equation Problems:

• Example: If you buy 3 notebooks for $15, how much does one notebook cost? • Translation to Equation: $$3x = 15$$ • Solution: Divide both sides by 3 to find $$x = 5$$. So, each notebook costs$5.
2. Problems Involving Brackets:

• Example: A rectangle has a length that is 3 units more than twice its width. If the perimeter is 24 units, what are the dimensions?
• Translation to Equation: Let ww be the width. The length is 2w+32w + 3. The perimeter equation is 2(w+2w+3)=242(w + 2w + 3) = 24.
• Solution: Simplify and solve 2(3w+3)=242(3w + 3) = 24, leading to 3w+3=123w + 3 = 12, then 3w=93w = 9, and finally w=3w = 3. The width is 3 units, and the length is 2(3)+3=92(3) + 3 = 9 units.
3. Problems Involving Fractions:

• Example: If you save 13\frac{1}{3} of your allowance every week and you saved $50 in 5 weeks, what is your weekly allowance? • Translation to Equation: Let aa be the weekly allowance. Each week, you save 13a\frac{1}{3}a. Over 5 weeks, 5⋅13a=505 \cdot \frac{1}{3}a = 50. • Solution: Simplify and solve 53a=50\frac{5}{3}a = 50, leading to 5a=1505a = 150, then a=30a = 30. The weekly allowance is$30.
4. Combination of Brackets and Fractions:

• Example: You have $100 to spend on t-shirts. Each t-shirt costs$x and you also have to pay a fixed shipping fee of 14\frac{1}{4} of the total cost. How many t-shirts can you buy if you spend all your money?
• Translation to Equation: Let nn be the number of t-shirts. The total cost is nxnx, and the shipping fee is 14nx\frac{1}{4}nx. The equation is nx+14nx=100nx + \frac{1}{4}nx = 100.
• Solution: Combine terms to get 54nx=100\frac{5}{4}nx = 100, then 5nx=4005nx = 400, leading to nx=80nx = 80. Solve for nn to find the number of t-shirts you can buy given the price xx.

## Translating verbal statements into simple linear equations

Translating verbal statements into simple linear equations involves converting a written description of a problem into an algebraic equation that can be solved. Here are the steps and examples to help understand the process