Equations

What is an Equation?

Think of an equation as a statement of balance. In algebra, this balance is denoted by an equals sign (=). An equation tells us that the quantity on the left-hand side is equal to the quantity on the right-hand side.

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Equations usually contain variables, which are symbols that stand in for unknown values. Often, these variables are denoted by letters such as $x$ , $y$ or $z$ .

The Importance of the Equals Sign

Before diving into solving equations, let’s first understand what the equals sign signifies.

When you see an equation like $x + 3 = 7$ , it is telling you that the expression “x $+ 3$ ” has the same value as 7. The objective is to determine what value of x makes this statement true.

Moving Variables Across the Equals Sign

When moving variables across the equals sign in an equation, it’s important to maintain the equality by performing the same operation on both sides of the equation. Here are the six key rules and concepts to keep in mind:

Rule 1: Addition and Subtraction

If you add or subtract a term from one side of the equation, you must do the same on the other side. For example, consider $2 \times x = 6$ . To isolate $x$ , you’ll subtract 2 from both sides to obtain $2 \times x - 2 = 4$ .

Rule 2: Multiplication and Division

Similar to addition and subtraction, if you multiply or divide one side of the equation, you must perform the same operation on the other side. If you have $3 \times x = 9$ , dividing both sides by 3 results in $x = 3$ .

Rule 3: Reverse Operations

When moving a term across the equals sign, use the reverse operation. For example, if you have $x + 3 = 4$ , the “+ 3” becomes “- 3” when moved across the equals sign. Therefore, you get $x = 1$ .

Rule 4: Distributive Property

If a term outside a bracket is being multiplied by multiple terms inside a bracket, you can distribute that term before moving variables across. For example, $2 \times (x + 3) = 10$ can be expanded as $2 \times x + 6 = 10$ first, before isolating $x$ .

Rule 5: Changing Sides Changes Signs

When you move a term from one side of the equation to the other, its sign changes. For example, $x - 4 = 7$ will turn into $x = 7 + 4$ or $x = 11$ when you move “- 4” to the other side.

Rule 6: Variables on Both Sides

When there are variables on both sides of the equation, try to get them all on one side by using the rules above. For example, in $2 \times x + 1 = 3 \times x - 2$ , you would move all terms involving x to one side to get $x = 3$ .

Let’s work through some examples

Examples

Example 1: Solve the equation $5 = x - 2$ .

To solve for $x$ , you would want to move the -2 from the right side to the left. This involves adding 2 to both sides of the equation. The equation then becomes $5 + 2 = x$ , simplifying to $7 = x$ or $x = 7$ .

Example 2: Solve this equation with variables on both sides: $3 \times x + 4 = 2 \times x + 7$ .

The goal is to collect all the $x$ terms on one side of the equation to solve it.

Start by subtracting $2 \times x$ from both sides, which gives you $x + 4 = 7$ . Now, isolate $x$ by subtracting 4 from both sides. The equation then simplifies to $x = 3$ .

Practice Questions

Question 1:

Solve $x + 3 = 7$

Solution

To find the value of $x$ , you would simply subtract 3 from both sides of the equation.

This would give you $x=4$ .

Question 2:

Solve $2 \times x + 4 = 10$

Solution

Start by subtracting 4 from both sides to get $2 \times x = 6$ .

After this, divide both sides by 2. The answer is $x = 3$ .

Question 3:

Solve $5 = 2 \times x - 3$

Solution

Add 3 to both sides to get $8 = 2 \times x$ .

Divide both sides by 2, and you’ll find $x = 4$ .

Question 4:

Solve $4 \times x - 7 = 3 \times x + 2$

Solution

Start by moving all $x$ terms to one side by subtracting $3 \times x$ from both sides. This results in $x - 7 = 2$ . Next, add 7 to both sides, leading to $x = 9$ .

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Equations

What is an Equation?

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The Importance of the Equals Sign

Moving Variables Across the Equals Sign

Rule 1: Addition and Subtraction

Rule 2: Multiplication and Division

Rule 3: Reverse Operations

Rule 4: Distributive Property

Rule 5: Changing Sides Changes Signs

Rule 6: Variables on Both Sides

Examples

Practice Questions

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