The equation “y = mx + c” represents a linear equation commonly used in mathematics to describe a straight line on a graph. In this article, we will explore various questions and answers related to this equation to help you understand its components and applications better.

In the equation “y = mx + c,” “y” represents the dependent variable (usually the output), “x” represents the independent variable (usually the input), “m” represents the slope of the line, and “c” represents the y-intercept (the point where the line crosses the y-axis).

Contents

- 1 2. Finding the Slope (m) and Y-Intercept (c)
- 2 3. Applications of the Linear Equation
- 3 4. Practice Questions and Answers
- 3.1 Example 1: y=mx+c form
- 3.2 Example 2: y=mx+c form
- 3.3 Example 3: rearrange the equation when x is the subject
- 3.4 Example 4: changing the subject including a fraction
- 3.5 Example 5: changing the subject including brackets
- 3.6 Example 6: changing the subject including decimals
- 3.7 Common misconceptions
- 3.8 Practice y=mx+c questions
- 3.9 y=mx+c GCSE questions
- 3.10 Example 1: sequence with a term to term rule of +3.
- 3.11 Example 2: sequence with a term to term rule of -1.

- 4 Geometric sequences examples
- 5 Quadratic sequences example
- 6 Sequence rule to find a term
- 7 How to find the nth term of a sequence
- 8 How to rearrange formula to change the subject of the formula
- 9 Rearranging equations examples
- 9.1 Example 1: multiple step but with single variable
- 9.2 Example 2: questions involving x2
- 9.3 Example 3: questions involving √x
- 9.4 Example 4: factorisation of the variable is required
- 9.5 Example 5: factorisation of the variable is required
- 9.6 Common misconceptions
- 9.7 Practice rearranging equations questions
- 9.8 Rearranging equations GCSE questions

## 2. Finding the Slope (m) and Y-Intercept (c)

To find the slope of a line represented by the equation “y = mx + c,” you can compare the coefficient of “x” (m) to determine the rise over run. The y-intercept (c) is the value of “y” when “x” is 0.

## 3. Applications of the Linear Equation

The equation “y = mx + c” is widely used in various fields, including physics, engineering, economics, and finance, to model relationships between two variables.

## 4. Practice Questions and Answers

- Question: Given the equation “y = 3x + 2,” find the slope (m) and y-intercept (c). Answer: The slope (m) is 3, and the y-intercept (c) is 2.
- Question: For the equation “y = -2x + 5,” determine the slope (m) and y-intercept (c). Answer: The slope (m) is -2, and the y-intercept (c) is 5.
- Question: If the equation of a line is “y = 0.5x – 1,” what is the slope (m) and y-intercept (c)? Answer: The slope (m) is 0.5, and the y-intercept (c) is -1.
- Question: Find the slope (m) and y-intercept (c) of the equation “y = 4x + 3.” Answer: The slope (m) is 4, and the y-intercept (c) is 3.
- Question: What is the slope (m) and y-intercept (c) of the linear equation “y = -1.5x – 2”? Answer: The slope (m) is -1.5, and the y-intercept (c) is -2.
- Question: Determine the slope (m) and y-intercept (c) of the line represented by the equation “y = 2x – 4.” Answer: The slope (m) is 2, and the y-intercept (c) is -4.

### Example 1: y=mx+c form

State the gradient and y-intercept of the line y = −3x + 8.

**Rearrange the equation to make y the subject**

The equation y = −3x + 8 is already in the general form of y=mx+c so we can progress to step 2 straight away.

2**Substitute x = 0 into the equation to find the y-intercept**

When x = 0,

The y-intercept of the line y = −3x + 8 is 8, or c = 8.

3**State the coefficient of x (the gradient)**

The coefficient of x is −3.

The gradient of the line y = −3x + 8 is −3, or m = −3.

Solution: m = −3, c = 8

### Example 2: y=mx+c form

State the gradient and y-intercept of the line y = 7 − x.

**Rearrange the equation to make y the subject**

**Substitute x = 0 into the equation to find the y-intercept**

**State the coefficient of x (the gradient)**

### Example 3: rearrange the equation when x is the subject

State the gradient and y-intercept of the line x = y + 10.

**Rearrange the equation to make y the subject**

**Substitute x = 0 into the equation to find the y-intercept**

**State the coefficient of x (the gradient)**

### Example 4: changing the subject including a fraction

State the gradient and y-intercept of the line 2x = 6y − 15.

**Rearrange the equation to make y the subject**

**Substitute x = 0 into the equation to find the y-intercept**

**State the coefficient of x (the gradient)**

### Example 5: changing the subject including brackets

State the gradient and y-intercept of the line 3x = 4(y − 5).

**Rearrange the equation to make y the subject**

**Substitute x = 0 into the equation to find the y-intercept**

**State the coefficient of x (the gradient)**

### Example 6: changing the subject including decimals

State the gradient and y-intercept of the line

**Rearrange the equation to make y the subject**

**Substitute x = 0 into the equation to find the y-intercept**

**State the coefficient of x (the gradient)**

### Common misconceptions

**Incorrect inverse operation**

When rearranging equations, instead of applying the inverse operation to the value being moved, the value is simply moved to the other side of the equals sign.

E.g.

y + 5 = 2x is rearranged to make y = 2x + 5 ✘

**Stating the value of m and c**

A common error is to incorrectly state the values of m and c as a result of not rearranging the equation so that it is in form of y=mx+c.

Take example 3 above when x = y + 10. The gradient would be stated as 1 as this is the coefficient of x however the value of the y-intercept would be incorrectly stated as **10.**

The correct answer for the y intercept is −10 as we can rearrange the equation to make y the subject. See below.

**Mixing up the gradient and the y-intercept**

Take, for example, the equation y = 10 + 3x. Here, the coefficient of x is 3 and the value of y when x = 0 is 10. This means that the gradient is 3 and the y-intercept is 10, not the other way around.

### Practice y=mx+c questions

1. State the gradient, $m$, and $y$-intercept, $c$, for the equation

$y=−5x+9$

2. State the gradient, $m$, and $y$-intercept, $c$, for the equation

$y=6−x$

3. State the gradient, $m$, and $y$-intercept, $c$, for the equation

$x=2y+5$

4. State the gradient, $m$, and $y$-intercept, $c$, for the equation

$3x=5y−6$

5. State the gradient, $m$, and $y$-intercept, $c$, for the straight line

$2x=3(3+y)$

6. State the gradient, $m$, and $y$-intercept, $c$, for the equation of the line

$0.5x+0.75y=0.25$

### y=mx+c GCSE questions

1. Given that the coordinate $(3,4)$ lies on the line $y=3x+c$ calculate the $y$-intercept of the straight line.

**(2 Marks)**

2. (a) The coordinate $A=(0,2)$ lies on a straight line. The gradient of the line is $5$. Using this information, state the equation of the straight line.

(b) Write the equation of a line that is parallel to a) in the form $y=mx+c$.

**(4 Marks)**

3. Show $m=2$ for the straight line $8x−4y=12.$

**(3 Marks)**

### Example 1: sequence with a term to term rule of +3.

We add three to the first term to give the next term in the sequence, and then repeat this to generate the sequence.

### Example 2: sequence with a term to term rule of -1.

We subtract 1 from the first term to give the next term in the sequence, and then repeat this to generate the sequence.

We can work out previous terms by doing the opposite of the term to term rule.

**2Geometric sequences**

A geometric sequence is an ordered set of numbers that progresses by multiplying or dividing each term by a common ratio.

If we **multiply **or** divide **by the **same **number each time to make the sequence, it is a **geometric sequence**.

**Step-by-step guide:** Geometric sequences

## Geometric sequences examples

### Example 3: sequence with a term to term rule of ×2.

We multiply the first term by 2 to give the next term in the sequence, and then repeat this to generate the sequence.

### Example 4: sequence with a term to term rule of ÷2.

We divide the first term by 2 to give the next term in the sequence, and then repeat this to generate the sequence.

**3Quadratic sequences**

A quadratic sequence is an ordered set of numbers that follow a rule based on the sequence n^{2} = 1, 4, 9, 16, 25, …(the square numbers).

The difference between each term is not equal, but the second difference is.

**Step-by-step guide:** Quadratic sequences

## Quadratic sequences example

### Example 5: sequence with common second difference of +2.

We find the first difference of the sequence and then find the term to term rule for the second difference. The second difference will always be the same for quadratic sequences.

### Example 6: square numbers

A square number is the result when a number is multiplied by itself.

E.g. 1×1=1, 2×2=4, 3×3=9 *etc*.

The square numbers can form a sequence: 1, 4, 9, 16, 25, 36, 49…

nth Term = n^{2}

### Example 7: cube numbers

A cube number is the result when a number is multiplied by itself three times.

E.g.

1×1×1=1, 2×2×2=8, 3×3×3=27 *etc. *

The cube numbers can form a sequence: 1, 8, 27, 64, 125…

nth Term = n^{3}

### Example 8: triangular numbers

The triangular numbers as numbers that can form a triangular dot pattern. They are also special type of quadratic sequence.

We can generate a sequence of triangular numbers by adding one more to the term to term rule each time:

### Example 9: Fibonacci numbers

We can generate the Fibonacci Sequence of numbers by adding the previous two numbers together to work out the next term.

First and second terms:

We start with 0, 1

0+1=1, so the third term is 1.

**Sequence**: 0, 1, 1

Fourth Term:

1+1=2

**Sequence**: 0, 1, 1, 2

Fifth Term:

1+2=3

**Sequence**: 0, 1, 1, 2, 3

We can continue to follow the pattern to generate an infinite sequence.

**The Fibonacci Sequence:**

**0, 1, 1, 2, 3, 5, 8, 13, 21, …**

The Fibonacci Sequence forms a spiral that is seen throughout nature.

## Sequence rule to find a term

We use the nth term of a sequence to work out a particular term in a sequence.

By substituting in the number of the term we want to find as ‘n’ we can generate the specific term in the sequence.

E.g.

What is the nth term and the 21^{st} term of this sequence?

4, 10, 16, 22, …

The nth term of this sequence is 6n – 2.

To find the 21^{st} term, n = 21

(6 x 21) – 2 = 124

The 21^{st} term is 124.

E.g.

Given the nth term rule, 3n + 4, find the 20^{th} and 100^{th}term for this sequence.

To find the 20^{th} term, n = 20

(3 x 20) + 4 = 64

To find the 100^{th} term, n = 100

(3 x 100) + 4 = 304. The 20^{th} term is 64 and the 100^{th} term is 304.

## How to find the nth term of a sequence

The n^{th }term is a formula that enables us to find any term in a sequence.

We will need to be able to find the **nth term of a linear (arithmetic) sequence**, and the **nth term of a quadratic sequence**.

We can make a sequence using the nth term by substituting different values for the term number n into it.

## How to rearrange formula to change the subject of the formula

In order to do rearrange formula to change the subject of the formula, I need to follow the steps:

- Identify the variable you need to make the subject of the formula.
- Isolate the variable – this step may look slightly different depending on the format of the question.

– Remove any fractions by multiplying by the denominator/s

– Divide by the coefficient of the variable

– Square root or square both sides of the equation

***not always required*** - Rearrange the equation so each term containing the term you want to be the subject is on one side of the equation – normally the left-hand side.
- Factorisation may be needed if you have multiple different terms containing your subject e.g. factorise 2x + 3xy to x(2 + 3y)

**not always required** - Perform an operation to ensure only the single variable is left as the subject.

### How to rearrange formula to change the subject of the formula

## Rearranging equations examples

### Example 1: multiple step but with single variable

p = 2(x − 3)

- Identify the variable to be made the subject.

In this question it is x.

2 Divide each side of the equation by 2

3 Add 3 to each side of the equation

**Answer:**

**Fully worked out answer:**

### Example 2: questions involving x^{2}

Identify the variable to be made the subject.

Add 4 to each side of the equation.

Square root each side. Remember: the square root can be a + or -.

### Example 3: questions involving √x

Identify the variable to be made the subject.

Subtract ‘n’ from each side of the equation.

The inverse operation of ‘square root’ is to ‘square’ each side.

Divide each side by the equation by the ‘coefficient of $x$‘. Here the coefficient is 3. *Note in this question step 4 was not required.*

### Example 4: factorisation of the variable is required

Identify the variable to be made the subject.

Multiply each side of the equation by the denominator.

Expand the bracket on the left hand side of the equation. This will help us get all terms with $x$ onto one side of the equation.

If we factorise the left side of the equation we will be left with only one of the variable $x$.

Divide by ($y−2z$). This will leave $x$ as the subject of the equation.

### Example 5: factorisation of the variable is required

Identify the variable to be made the subject.

As in the previous example, we are multiplying the equation by the denominator. In this example we have denominators on both sides we multiply by both.

Expand the bracket on the LHS and RHS of the equation. This will help to get all terms with $x$ onto one side of the equation.

If we factorise the left side of the equation we will be left with only one of the variable $x$.

Now divide by (*a+21)*. This will leave $x$ as the subject of the equation.

### Common misconceptions

**When we perform an operation to one side of the equation we have to do to the other.****Incorrect use of the inverse operation**.**Incorrectly following the order of operations.****All variables of the subject need to be on one side of the equal sign.**

**When we square root a number/variable the answer can be positive or negative.**

√x should be written as ± √x

**Not factorising when we have the subject in more than one term.**

E.g. Make x the subject

### Practice rearranging equations questions

1. Make $a$ the subject of the formula $h=3(a+7)$

2. Make $b$ the subject of the formula $p=b_{2}−9k$

3. Make $c$ the subject of the formula $g=c−r $

4. Make $d$ the subject of the formula $y=dd+ $

5. Make $e$ the subject of the formula $3q =e+−e $

6. Make $f$ the subject of the formula $5l =f+l−f $

### Rearranging equations GCSE questions

1. Make *x* the subject of the formula

(2 marks)

2. Make *s* the subject of

(2 marks)

3. Make *g* the subject of the formula

The equation “y = mx + c” is a fundamental concept in algebra, representing a linear relationship between two variables. Understanding its components, such as the slope (m) and y-intercept (c), is crucial in various fields of study and real-world applications. By practicing questions and answers related to this equation, you can strengthen your grasp of linear relationships and their significance in mathematics and beyond.