Solving Linear Equations

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:

1. Equations with one unknown (like x). $(2x+5=11)$
2. Equations that have unknowns on both sides. $(3x +4=2x+7)$
3. Equations that include brackets. $(2(x-3)=8)$
4. 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.

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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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The general form of a linear equation looks like this:

$ax + by + c = 0$

Or sometimes, it's written as:

$y = ax + b$

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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.

How to "Undo" Operations in an Equation:

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.

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a) Solving Linear Equations with One Unknown

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

5. Rearrange the equation so that the unknown variable (x) is by itself on one side.
6. 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.

10. 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

11. 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!

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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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b) Solving Linear Equations with an Unknown on Both Sides

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

7. Get all the letters (variables) on the same side of the equation.
8. 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.)
9. 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.

13. 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:

$3x = 3$

14. 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:

$x = 1$

15. 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!

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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:

10. Expand the brackets to remove them and simplify the equation.
11. 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.

d) Solving Linear Equations with Fractions

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

12. 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.
13. Expand the brackets to simplify the equation.
14. Move all the letters (variables) to the same side of the equation.
15. 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.

22. 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:

$x = 6$

23. 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!

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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.

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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 $(ax + b = c)$, where a, b, and c are numbers.

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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.

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Types of Linear Equations and How to Solve Them:

24. Equations with One Unknown:

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

25. 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.

26. Equations with Brackets:

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

27. 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 $x$.

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Key Tips:

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

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