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📐 Mathematics - Class 7
Lesson 24 · Equations, Inequalities and Line Graphs

Linear Equations in One Variable

MathematicsSubject
14 minEstimated read
An equation is a balance

Whatever you do to one side you must do to the other. That single rule is the whole method.

An expression and an equation are different

3x + 5 is an expression with a value for every x. 3x + 5 = 20 is a claim that can be true or false.

Undo the operations in reverse order

Take away the 5 first, then divide by the 3. The last thing done is the first thing undone.

Step Equation What was done to both sides
13x + 5 = 20The equation as given
23x = 15Subtracted 5 from both sides
3x = 5Divided both sides by 3
4Check: 3(5) + 5 = 20Substitute back into the original
Do it to the whole side, not part of it

If 3x + 6 = 15 and you divide by 3, you get x + 2 = 5, not x + 6 = 5.

Letters on both sides

Collect all the x terms on one side and all the numbers on the other, using the same balance rule.

Always check by substituting back

For x = 4 in 5x − 3 = 2x + 9, the left is 17 and the right is 17. This check catches every arithmetic slip.

Turning a problem into an equation

Name the unknown first and say what it stands for, then write the sentence in symbols.

A solution is one number, not a rearrangement

2x = 8 is not an answer, because x has not been found yet. Read your last line and ask whether it gives the value of x.

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Worked Example 1

Solve 4x − 7 = 21.

Read what was done to x×4 first, then −7
Undo the subtraction4x = 28
Undo the multiplicationx = 7
Check4(7) − 7 = 21 ✓
Worked Example 2

Solve 5x − 3 = 2x + 9.

Get the x terms together3x − 3 = 9
Now it is the familiar shape3x = 12
Finish the jobx = 4
Check both sides separatelyLeft 17, right 17 ✓
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Activity 24.1

The envelope balance

What you needTwo envelopes, identical counters such as bottle caps, and a sheet with a line down the middle.
Set it upBuild an equation: three envelopes and five counters on the left, twenty counters on the right.
Now solve it physicallyRemove five counters from each side, split the rest into three equal groups, then open an envelope.
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1

An equation is a balance: the two sides weigh exactly the same.

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1What is the difference between an expression and an equation?
2Why must you do the same thing to both sides of an equation?
3Why do you undo the operations in reverse order?
4How do you solve an equation with x on both sides?
5How do you turn a word problem into an equation?

Question 1 of 10

1An equation differs from an expression because it
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Slide 1 of 4
Unit 24 · Linear Equations

Linear Equations

Mathematics · Grade 7

What You Will Be Able To Do

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Solve an equation by keeping both sides in balance at every step

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Undo operations in the right order, and know why order matters

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Handle equations with x on both sides of the equals sign

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Turn a problem written in words into an equation and answer it

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The equals sign says that the two sides weigh exactly the same. So whatever you do to one side you must do to the other.

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Presenter notes: Open by holding a sealed envelope of counters and telling the class there is a fixed number inside that you are not going to reveal. Say that three of these envelopes plus five loose counters weigh the same as twenty counters, and ask how many are inside. Somebody will work it out. Point out that they have just solved an equation, and that today is about doing the same thing on paper.