## Solving Linear Equations

Learning level: Grade school

## 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

**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.

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**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 $w$ be the width. The length is $2w+3$. The perimeter equation is $2(w+2w+3)=24$.**Solution**: Simplify and solve $2(3w+3)=24$, leading to $3w+3=12$, then $3w=9$, and finally $w=3$. The width is 3 units, and the length is $2(3)+3=9$ units.

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**Problems Involving Fractions**:- Example:
*If you save $31 $ of your allowance every week and you saved $50 in 5 weeks, what is your weekly allowance?***Translation to Equation**: Let $a$ be the weekly allowance. Each week, you save $31 a$. Over 5 weeks, $5⋅31 a=50$.**Solution**: Simplify and solve $35 a=50$, leading to $5a=150$, then $a=30$. The weekly allowance is $30.

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**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 $41 $ of the total cost. How many t-shirts can you buy if you spend all your money?***Translation to Equation**: Let $n$ be the number of t-shirts. The total cost is $nx$, and the shipping fee is $41 nx$. The equation is $nx+41 nx=100$.**Solution**: Combine terms to get $45 nx=100$, then $5nx=400$, leading to $nx=80$. Solve for $n$ to find the number of t-shirts you can buy given the price $x$.

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## 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