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).

Contents

1 2. Finding the Slope (m) and Y-Intercept (c)
2 3. Applications of the Linear Equation
3 4. Practice Questions and Answers
4 Geometric sequences examples
- 4.1 Example 3: sequence with a term to term rule of ×2.
- 4.2 Example 4: sequence with a term to term rule of ÷2.
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
- 8.1 How to rearrange formula to change the subject of the formula
9 Rearranging equations examples

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.

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

When x = 0,

\begin{matrix} y = (- 3 \times 0) + 8 y = 0 + 8 y = 8 \end{matrix}

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.

Rearrange the equation to make y the subject

Show step

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

Show step

State the coefficient of x (the gradient)

Show step

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

Show step

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

Show step

State the coefficient of x (the gradient)

Show step

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

Show step

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

Show step

State the coefficient of x (the gradient)

Show step

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

Show step

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

Show step

State the coefficient of x (the gradient)

Show step

Example 6: changing the subject including decimals

State the gradient and y-intercept of the line

x = \frac{y + 0.85}{0.2} .

Rearrange the equation to make y the subject

Show step

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

Show step

State the coefficient of x (the gradient)

Show step

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

m = - 5, c = 9

m = 9, c = 5

m = 9, c = - 5

m = - 5, c = 4

m = 6, c = - 1

m = - 1, c = 6

m = 5, c = 6

m = 0, c = 5

m = 2 1, c = - 2 5

m = 2, c = 5

m = - 2, c = - 5

m = 1, c = 7

m = 5 3, c = 5 6

m = 3, c = - 6

m = 2, c = - 1

m = 2 y, c = 5

m = 3 2, c = - 3

m = 2, c = 0

m = 6, c = 9

m = 3 2, c = 3

m = - 3 2, c = 3 1

m = 0.5, c = 0.5

m = 0.75, c = 0.5

m = 1.5, c = 0.5

y=mx+c GCSE questions

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

(2 Marks)

Show answer

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 $y = m x + c$ .

(4 Marks)

Show answer

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

(3 Marks)

Show answer

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 n² = 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 = 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:

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^thterm 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^thterm 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.

p = 2 (x - 3)

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:

\frac{p}{2} + 3 = x

Fully worked out answer:

Example 2: questions involving x²

y = x^{2} - 4

Identify the variable to be made the subject.

Show step

Add 4 to each side of the equation.

Show step

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

Show step

Example 3: questions involving √x

y = \sqrt{3 x} + n

Identify the variable to be made the subject.

Show step

Subtract ‘n’ from each side of the equation.

Show step

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

Show step

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.

Show step

Example 4: factorisation of the variable is required

y = \frac{2 x z}{x - 5}

Identify the variable to be made the subject.

Show step

Multiply each side of the equation by the denominator.

Show step

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.

Show step

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

Show step

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

Show step

Example 5: factorisation of the variable is required

\frac{a}{3} = \frac{2 - 7 x}{x - 5}

Identify the variable to be made the subject.

Show step

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.

Show step

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.

Show step

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

Show step

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

Show step

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.

E . g . x = 2 x + 2

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

E . g . \sqrt{4} = \pm 2

√x should be written as ± √x

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

E.g. Make x the subject

\begin{matrix} 2 x + 3 x y & = 3 y x (2 + 3 y) & = 3 y x & = \frac{3 y}{2 + 3 y} \end{matrix}

Practice rearranging equations questions

h - 7 = a

3 h - 7 = a

3 h - 21 = a

h - 21 = a

p + 9 k = b

\pm p + 9 k = b

\pm p - 9 k = b

\pm 9 k - p = b

g^{2} + r = c

5 g ^{2} - r = c

5 g ^{2} + r = c

g^{2} - r = c

d = 4 y 1 - 3

d = 4 y - 3 1

d = 4 y + 3 1

d = 4 y 1 + 3

e = q - 6 18 - q

e = q + 6 18 - q

e = q + 6 18 + q

e = 6 18 - q

f = 2 l ^{3} + 15 20 l + 9 l ^{3}

f = 2 l ^{3} + 15 20 l - 9 l ^{3}

f = 2 l ^{3} - 15 20 l - 9 l ^{3}

f = 2 l ^{3} - 15 20 l + 9 l ^{3}

Rearranging equations GCSE questions

1. Make x the subject of the formula

$y = 2 x + 4$

Show answer

(2 marks)

2. Make s the subject of

$v^{2} = u^{2} + 2 a s$

Show answer

(2 marks)

3. Make g the subject of the formula

$y = \sqrt{\frac{g + 6}{5}}$

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.

Y = mx + c Questions and Answers

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

3. Applications of the Linear Equation

4. Practice Questions and Answers

Example 1: y=mx+c form

Example 2: y=mx+c form

Example 3: rearrange the equation when x is the subject

Example 4: changing the subject including a fraction

Example 5: changing the subject including brackets

Example 6: changing the subject including decimals

Common misconceptions

Practice y=mx+c questions

y=mx+c GCSE questions

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

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

Geometric sequences examples

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

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

Quadratic sequences example

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

Example 6: square numbers

Example 7: cube numbers

Example 8: triangular numbers

Example 9: Fibonacci numbers

Sequence rule to find a term

How to find the nth term of a sequence

How to rearrange formula to change the subject of the formula

How to rearrange formula to change the subject of the formula

Rearranging equations examples

Example 1: multiple step but with single variable

Example 2: questions involving x²

Example 3: questions involving √x

Example 4: factorisation of the variable is required

Example 5: factorisation of the variable is required

Common misconceptions

Practice rearranging equations questions

Rearranging equations GCSE questions

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

3. Applications of the Linear Equation

4. Practice Questions and Answers

Example 1: y=mx+c form

Example 2: y=mx+c form

Example 3: rearrange the equation when x is the subject

Example 4: changing the subject including a fraction

Example 5: changing the subject including brackets

Example 6: changing the subject including decimals

Common misconceptions

Practice y=mx+c questions

y=mx+c GCSE questions

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

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

Geometric sequences examples

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

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

Quadratic sequences example

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

Example 6: square numbers

Example 7: cube numbers

Example 8: triangular numbers

Example 9: Fibonacci numbers

Sequence rule to find a term

How to find the nth term of a sequence

How to rearrange formula to change the subject of the formula

How to rearrange formula to change the subject of the formula

Rearranging equations examples

Example 1: multiple step but with single variable

Example 2: questions involving x2

Example 3: questions involving √x

Example 4: factorisation of the variable is required

Example 5: factorisation of the variable is required

Common misconceptions

Practice rearranging equations questions

Rearranging equations GCSE questions

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Example 2: questions involving x²