🇨🇩 Algebra: expressions and equations

Linear equations in D

⏱ 25 minOfficial programme: CD-E7-ALG-05

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🎯 In this lesson, you will learn to

Solve one-variable linear equations incl. fractional coefficients.
Model commerce, health, environment problems with equations.

The lesson

Introduction

Mbuyi buys 3 exercise books and a pen at $800$ FC; he pays $5\,300$ FC in all. What is the price of one exercise book? If we call this price $x$ , we can write $3x+800=5\,300$ : this is an equation. In this lesson you learn to solve first-degree equations in one unknown, even with decimals or fractions, and to turn a problem into an equation.

Explanation

Equation and solution

A first-degree equation in one unknown can be written in the form $ax+b=c$ , where $x$ is the unknown and $a\ne0$ . A solution is a value of $x$ that makes the equality true. Here we work in $\mathbb{D}$ , the set of decimal numbers.

Equivalence rules (they give an equation with the same solutions):

you may add or subtract the same number on both sides;
you may multiply or divide both sides by the same non-zero number.

Practical consequence: a term that moves to the other side changes sign.

Explanation

Solving method

Remove the brackets (expand).
If the equation contains fractions, multiply both sides by a common denominator to clear them.
Put the terms in $x$ on one side and the numbers on the other.
Simplify, then divide by the coefficient of $x$ .
Check by replacing $x$ with the value found in the original equation.

Turning a problem into an equation: choose the unknown (and its unit), translate the problem into an equality, solve, check, then answer with a sentence.

Worked example

Problem: Solve $5x-7=2x+8$ , then $0.4x+1.2=3$ .

Step 1: Subtract $2x$ from both sides: $3x-7=8$ .

Step 2: Add $7$ : $3x=15$ , so $x=5$ . Check: $5\times5-7=18$ and $2\times5+8=18$ .

Step 3: For the second one: $0.4x=3-1.2=1.8$ .

Step 4: $x=1.8\div0.4=18\div4=4.5$ . Check: $0.4\times4.5+1.2=1.8+1.2=3$ .

Answer: $x=5$ for the first equation; $x=4.5$ for the second one.

Worked example

Problem: Solve $\dfrac{x}{3}+\dfrac{x}{4}=\dfrac{7}{2}$ .

Step 1: Common denominator of $3$ , $4$ and $2$ : $12$ . Multiply both sides by $12$ : $4x+3x=42$ .

Step 2: $7x=42$ , so $x=6$ .

Step 3: Check: $\dfrac{6}{3}+\dfrac{6}{4}=2+1.5=3.5=\dfrac{7}{2}$ .

Answer: $x=6$ .

Worked example

Problem: Mbuyi buys $3$ identical exercise books and a pen at $800$ FC. He pays $5\,300$ FC. What is the price of one exercise book?

Step 1: Unknown: $x$ , the price of one exercise book in FC.

Step 2: Equation: $3x+800=5\,300$ .

Step 3: $3x=4\,500$ , so $x=1\,500$ .

Step 4: Check: $3\times1\,500+800=5\,300$ .

Answer: one exercise book costs $1\,500$ FC.

Watch out

Watch out: when you multiply by a common denominator, you must multiply every term on both sides, even those without a denominator. In $\dfrac{x}{2}+1=4$ , multiplying by $2$ gives $x+2=8$ , not $x+1=8$ . Likewise, a $-$ sign before a bracket changes every sign inside.

Key points

You may add, subtract, multiply or divide (by a non-zero number) both sides by the same number.
A term that moves to the other side changes sign.
Fractions: multiply every term by a common denominator.
Always check the solution in the original equation.
Word problem: unknown, equation, solving, checking, answer sentence.

✍️ Exercises

Answer one exercise at a time. Your answer is checked right away, with an explanation.

Exercise 1 of 8

★ Easy

Solve the equation $x+7.5=12$ . Give the value of $x$ .

🎯 In this lesson, you will learn to

The lesson

👋 Introduction

📘 Explanation

📘 Explanation

✏️ Worked example

✏️ Worked example

✏️ Worked example

⚠️ Watch out

⭐ Key points

✍️ Exercises

Introduction

Explanation

Explanation

Worked example

Worked example

Worked example

Watch out

Key points