# Y = mx + c Questions and Answers

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).

## 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

1. 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.
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.
3. 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.
4. 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.
5. 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.
6. 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.

1. 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.

2Substitute 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.

3State 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.

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

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

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

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

### Example 5: changing the subject including brackets

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

### Example 6: changing the subject including decimals

State the gradient and y-intercept of the line

### 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, , and -intercept, , for the equation

2. State the gradient, , and -intercept, , for the equation

3. State the gradient, , and -intercept, , for the equation

4. State the gradient, , and -intercept, , for the equation

5. State the gradient, , and -intercept, , for the straight line

6. State the gradient, , and -intercept, , for the equation of the line

### y=mx+c GCSE questions

1.  Given that the coordinate lies on the line calculate the -intercept of the straight line.

(2 Marks)

2.  (a)  The coordinate lies on a straight line. The gradient of the line is . 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 .

(4 Marks)

3.  Show for the straight line

(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.

A quadratic sequence is an ordered set of numbers that follow a rule based on the sequence n2 = 1, 4, 9, 16, 25, …(the square numbers).

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

### 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 = n2

### 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 = n3

### 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:

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 21st term of this sequence?

4, 10, 16, 22, …

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

To find the 21st term, n = 21

(6 x 21) – 2 = 124

The 21st term is 124.

E.g.

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

To find the 20th term, n = 20

(3 x 20) + 4 = 64

To find the 100th term, n = 100

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

## How to find the nth term of a sequence

The nth 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:

1. Identify the variable you need to make the subject of the formula.
2. 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*
3. 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.
4. 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*
5. Perform an operation to ensure only the single variable is left as the subject.

## Rearranging equations examples

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

p = 2(x − 3)

1. 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

### 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 the subject of the formula

2. Make the subject of the formula

3. Make the subject of the formula

4. Make the subject of the formula

5. Make the subject of the formula

6. Make the subject of the formula

### 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.