Solving Linear Equations Revision Notes for Junior Cert Mathematics

Solving Linear Equations

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Learning intentions:

What is a linear equation?
Equations with one unknown (like x). $(2x+5=11)$
Equations that have unknowns on both sides. $(3x +4=2x+7)$
Equations that include brackets. $(2(x-3)=8)$
Equations that have fractions. $(\frac{x}{2} + 3 = 7)$

Linear Equations

In this section, we will learn how to solve simple equations called linear equations.

These are equations where the highest power of the variable (like x or y) is just 1.

We will cover:

Equations with one unknown (like x). $(2x+5=11)$
Equations that have unknowns on both sides. $(3x +4=2x+7)$
Equations that include brackets. $(2(x-3)=8)$
Equations that have fractions. $(\frac{x}{2} + 3 = 7)$ Remember: Don't worry if some of these terms seem confusing right now! We'll go through everything step by step.

What is a Linear Equation?

A linear equation is a type of equation where the variables (like x or y) are not multiplied by themselves.
In other words, they are not squared ( $x²$ ) or not cubed ( $x³$ ).
Instead, the variables have an exponent of 1, which we usually don't write. ( $x^1$ , written as $x$ )
Linear equation have an equals sign (=), which means that whatever is on the left side is exactly the same as what is on the right side.

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Why is it called "linear"? It's called a linear equation because if you were to draw it on a graph, the result would be a straight line.

The general form of a linear equation looks like this:

$ax + by + c = 0$

Or sometimes, it's written as:

$y = axe + b$

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Let's look at some examples: 6. $2x - 3y + 4 = 0$ Explanation: This is a linear equation because both $x$ and $y$ have an exponent of 1 (even though we don't write it). The equation has an equal sign, which indicates that the value of the expression on the left side is equal to the value on the right side, making it a valid equation. 7. $y = 5x + 2$ Explanation**:** This equation is also linear because y and x both have an exponent of 1. It's in the form of y = axe + b, which is another way to write a linear equation.

(The equation has an equal sign, which indicates that the value of the expression on the left side is equal to the value on the right side, making it a valid equation.)

$x^2 + y = 5$ Explanation: This is not a linear equation because $x$ is squared.

How to Solve Linear Equations

When we solve a linear equation, we're trying to find out what number makes the equation true.

We do this by simplifying the equation, and then doing the opposite of what the operation tells us to do.

Numbers attached to letters (like 2x) must stay joined together until you use a maths technique, like subtracting or dividing, to separate them. You can't just split them apart until you've used the correct step to do so.

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Key Tip: Always try to leave the variable (like x or y) alone for as long as you can. Focus on getting rid of the other numbers first.

How to "Undo" Operations in an Equation:

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If the equation has addition: To undo it, you need to subtract. 3x + 6 = 18

Subtract 6 from both sides:

3x + 6 -6= 18 -6

3x = 12

If the equation has subtraction: To undo it, you need to add.

3x - 6 = 18

Add 6 to both sides:

3x - 6 +6= 18 +6

3x = 24

If the equation has multiplication: To undo it, you need to divide.

3x = 12

Divide both sides by 3:

3x /3= 12 /3

x = 4

If the equation has division: To undo it, you need to multiply.

$\frac{x}{3} = 12$

Multiply both sides by 3:

$\frac{x}{3}$ (3) =12 (3)

x = 36

Important: After you've found the value of x (or whatever variable you're working with), it's a good idea to check your work. You can do this by substituting the value back into the original equation to make sure it's correct.

a) Solving Linear Equations with One Unknown

When solving a linear equation with one unknown (like x), follow these simple steps:

Rearrange the equation so that the unknown variable (x) is by itself on one side.
Find the value of the unknown variable (x) by doing the opposite of what the equation tells you to do. Let's break this down with an example.

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Example: Solving $(3x + 6 = 18)$ To solve this equation, we need to figure out what x is. We do this in two steps:

Step 1: Rearrange the equation. We want x to be by itself on one side of the equation.

Right now, x is being added to 6. The opposite of adding 6 is subtracting 6.

So, we subtract 6 from both sides of the equation:

3x + 6 = 18 Subtract 6 from both sides:

$3x + 6 \, (-6) = 18 \, (-6)$

$3x = 12$

Top Tip: Remember, "Letters to the left, numbers to the right." This means we try to get all the x terms on one side (usually the left side) and the numbers on the other side.

Step 2: Solve for x. Now, $3x$ means that 3 is multiplied by x.

To get x by itself, we need to do the opposite of multiplying by 3, which is dividing by 3.

$3x = 12$

Divide both sides by 3:

$\frac{3x}{3} = \frac{12}{3}$

x = 4

Check Your Work: It's always a good idea to check that your solution is correct. We do this by putting the value of x back into the original equation to see if it works:

Original equation: $(3x + 6 = 18)$

Substitute x = 4:

$3(4) + 6 = 18$

$12+6=18$

$18=18$

Since both sides are equal, our solution $x = 4$ is correct!

Summary of the Example:

Original equation: $(3x + 6 = 18)$
Step 1: Subtract 6 from both sides: $(3x = 12)$
Step 2: Divide by 3: $(x = 4)$
Check: Substitute x = 4 back into the original equation to confirm.

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Remember: "Letters to the left, numbers to the right." This simple rule helps you keep your work organised and makes it easier to solve the equation.

b) Solving Linear Equations with an Unknown on Both Sides

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How Is This Different from the First Type of Linear Equation (Type a)? In Type a, the unknown (like x) is only on one side of the equation. Here, the unknown appears on both sides, so we need an extra step to move all the x terms to the same side before solving. This makes it a bit more complex.

4x + 7 = 2x + 5

Note the variable on both sides: 4x on the left of the equals sign, 2x on the right of the equals sign

When solving a linear equation where the unknown (like x) appears on both sides, here's what you need to do:

Get all the letters (variables) on the same side of the equation.
Get the letters (variables) on one side and the numbers on the other. (Move the numbers to the right hand side, leaving the variables on the left.)
Find the value of the unknown by doing the opposite of what the equation tells you. Let's look at an example to see how this works.

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Example: Solving $(5x + 6 = 2x + 9)$ To solve this equation, we need to figure out what x is. We'll do this step by step:

Step 1: Get all the letters (variables) on the same side. We want all the x terms on one side. To do this, we need to move $(2x)$ to the left side of the equation. We do this by subtracting $(2x)$ from both sides.

$5x + 6 = 2x + 9$

Subtract $(2x)$ from both sides:

$5x (- 2x) + 6 = 2x (- 2x) + 9$

Which simplifies to:

$3x + 6 = 9$

Top Tip: Always try to eliminate the smaller variable first (in this case, $(2x)$ is smaller than $(5x)$ ).

Step 2: Get the numbers on the right side. Now that all the x terms are on the left side, we need to move the numbers to the right side. Here, we need to subtract 6 from both sides.

$3x + 6 = 9$

Subtract 6 from both sides:

$3x + 6 (- 6) = 9 (- 6)$

Which simplifies to:

$3x = 3$

Step 3: Solve for x. Now, 3x means that 3 is multiplied by x. To get x by itself, we need to do the opposite of multiplying by 3, which is dividing by 3.

$3x = 3$

Divide both sides by 3:

$\frac{3x}{3} = \frac{3}{3}$

Which simplifies to:

$x = 1$

Check Your Work: It's always important to check that your solution is correct. We do this by putting the value of x back into the original equation to make sure it works:

Original equation: $(5x + 6 = 2x + 9)$

Substitute x = 1:

$5(1) + 6 = 2(1) + 9$

Which simplifies to:

$5 + 6 = 2 + 9$

$11 = 11$

Since both sides are equal, our solution x = 1 is correct!

Summary of the Example:

Original equation: $(5x + 6 = 2x + 9)$
Step 1: Subtract $(2x)$ from both sides: $3x + 6 = 9$
Step 2: Subtract 6 from both sides: $3x = 3$
Step 3: Divide by 3: $x = 1$
Check: Substitute $x = 1$ back into the original equation to confirm.

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c) Solving Linear Equations with Brackets

When solving linear equations that have brackets, there are two main steps:

Expand the brackets to remove them and simplify the equation.
Solve the equation by rearranging it so that the unknown variable (like x) is by itself on one side. Then, do the opposite of what the equation tells you to do. Let's go through an example to see how this works.

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Example: Solving $4(x - 2) = 12$ To solve this equation, we need to figure out what x is. We'll do this step by step:

Step 1: Expand the brackets. Expanding the brackets means multiplying what's inside the brackets by the number outside. Here, we multiply everything inside the brackets by 4:

$4(x - 2) = 12$

Multiply 4 by both x and -2:

$4x - 8 = 12$

Step 2: Solve the equation by rearranging it. Now that we've expanded the brackets, we need to get x by itself on one side of the equation.

First, we get rid of (-8) by doing the opposite, which is adding 8 to both sides: $4x - 8 (+8) = 12 (+8)$

Which simplifies to:

$4x = 20$

Next, we solve for x by dividing both sides by 4 (since 4 is multiplying x): $\frac{4x}{4} = \frac{20}{4}$

Which simplifies to:

x = 5

Check Your Work: It's always important to check that your solution is correct. We do this by putting the value of x back into the original equation to see if it works:

Original equation: $4(x - 2) = 12$

Substitute x = 5:

$4(5 - 2) = 12$

$4(3) = 12$

$12=12$

Since both sides are equal, our solution x = 5 is correct!

Summary of the Example:

Original equation: $4(x - 2) = 12$
Step 1: Expand the brackets: $4x - 8 = 12$
Step 2: Add 8 to both sides: $4x - 8 (+8) = 12 (+8)$ Result: $4x = 20$
Step 3: Divide both sides by 4: $\frac{4x}{4} = \frac{20}{4}$

Result: $(x = 5)$

Check: Substitute x = 5 back into the original equation to confirm.

d) Solving Linear Equations with Fractions

When solving linear equations that include fractions, follow these four main steps:

Get rid of the fractions by multiplying each fraction by the denominator (the number on the bottom of the fraction) on the other side of the equals sign. Don't forget to include brackets around what you multiply.
Expand the brackets to simplify the equation.
Move all the letters (variables) to the same side of the equation.
Solve for the unknown variable by getting it by itself on one side, then doing the opposite of what the equation tells you to do. Let's go through an example to see how this works.

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Example: Solving $(\frac{5x - 2}{4} = \frac{2x + 2}{2}$ ) To solve this equation, we need to figure out what x is. We'll do this step by step:

Step 1: Get rid of the fractions. Fractions can make equations look tricky, but we can remove them by multiplying each fraction by the denominator on the other side of the equals sign.

$\frac{5x - 2}{4} = \frac{2x + 2}{2}$

Multiply each side by the opposite denominator (multiply the left side by 2 and the right side by 4):

$2(5x - 2) = 4(2x + 2)$

Step 2: Expand the brackets. Next, we need to simplify the equation by expanding the brackets (multiply what's inside the brackets by the number outside):

$2(5x - 2) = 4(2x + 2)$

This gives us:

$10x - 4 = 8x + 8$

Step 3: Move all the variables to the same side. Now, we need to get all the x terms on one side of the equation. To do this, subtract $(8x)$ from both sides:

$10x - 4 (-8x) = 8x + 8 (-8x)$

Subtract $(8x)$ from both sides:

$2x - 4 = 8$

Tip: Always subtract the smaller x term from the larger one to keep the variable positive.

Step 4: Solve for x. Now, we need to get x by itself on one side of the equation.

First, add 4 to both sides to eliminate $(-4)$ : $2x - 4 (+4) = 8 (+4)$

This simplifies to:

$2x = 12$

Next, divide both sides by 2 to solve for x: $\frac{2x}{2} = \frac{12}{2}$

Result:

$x = 6$

Check Your Work: Always check your solution by substituting x back into the original equation to see if it works:

Original equation: $(\frac{5x - 2}{4} = \frac{2x + 2}{2})$
Substitute $x = 6$ : $\frac{5(6) - 2}{4} = \frac{2(6) + 2}{2}$

Simplify both sides, by multiplying out the brackets:

5 x 6 = 30

2 x 6 = 12

$\frac{30 - 2}{4} = \frac{14}{2}$

$\frac{28}{4} = \frac{14}{2}$

$7 = 7$

Since both sides are equal, our solution x = 6 is correct!

Summary of the Example:

Original equation: $(\frac{5x - 2}{4} = \frac{2x + 2}{2})$
Step 1: Multiply each side by the opposite denominator: $2(5x - 2) = 4(2x + 2)$
Step 2: Expand the brackets: $10x - 4 = 8x + 8$
Step 3: Subtract $(8x)$ from both sides: $2x - 4 = 8$
Step 4: Add 4 to both sides: $2x - 4 (+4) = 8 (+4)$ Result: $(2x = 12)$
Step 5: Divide both sides by 2: $\frac{2x}{2} = \frac{12}{2}$ Result: $(x = 6)$
Check: Substitute x = 6 back into the original equation to confirm.

Summary: Solving Linear Equations

What is a Linear Equation?

A linear equation is a mathematical expression where the highest power of the variable (like $x$ or $y$ ) is $1$ .
It typically looks like $(axe + b = c)$ , where a, b, and c are numbers.

Basic Steps to Solve Any Linear Equation:

Rearrange the equation so that the variable is by itself on one side of the equation.
Do the opposite operation (like adding, subtracting, multiplying, or dividing) to isolate the variable.

Types of Linear Equations and How to Solve Them:

Equations with One Unknown:

Example: $(3x + 6 = 18)$
Steps: Subtract 6 from both sides, then divide by 3.

Equations with Unknowns on Both Sides:

Example: $(5x + 6 = 2x + 9)$
Steps: Move all $x$ terms to one side, then solve for $x$ by isolating it.

Equations with Brackets:

Example: $4(x - 2) = 12$
Steps: Expand the brackets, then solve for $x$ by isolating it.

Equations with Fractions:

Example: $(\frac{5x - 2}{4} = \frac{2x + 2}{2})$
Steps: Multiply both sides by the denominators to eliminate fractions, then solve for $x$ .

Key Tips:

Always perform the same operation on both sides of the equation.
Check your work by substituting the solution back into the original equation.

Solving Linear Equations (Junior Cert Mathematics): Revision Notes