How do I find the [katex] \textbf{x} [/katex]-intercept of a line?

The process for finding the [katex] x [/katex]-intercept of a line is similar to finding the [katex] y [/katex]-intercept. Instead of substituting [katex] x=0, [/katex] you would substitute [katex] y=0, [/katex] and solve the equation for [katex] x. [/katex]

How To Find Y Intercept - Math Steps, Examples & Questions

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How to find y intercept

Here you will learn about how to find the $y$ intercept from a straight line graph, including straight lines in the slope intercept form, $y=mx+b$ and standard form, $ax+by=c.$

Students will first learn about how to find the $y$ intercept in $8$ th grade math with their work with linear functions.

What is the y intercept and how do you find the y-intercept?

Finding the $y$ intercept of a straight line is an important skill used to solve algebraic and real-life problems involving straight line graphs.

The intercepts of a graph are where the graph crosses the coordinate axes.

You can find the $y$ intercepts of graphs of all types of functions including straight lines, quadratic functions, cubic functions and others. In this case, the $y$ intercept is the point where the function crosses or intercepts the $y$ -axis.

To find the $y$ intercept(s) of a line:

Substitute $x=0$ into the equation of the function and evaluate for $y.$

To find the $x$ intercept(s) of a line:

Substitute $y=0$ into the equation of the function and evaluate for $x.$

Note, this is a useful strategy to draw the equation of a straight line.

Step-by-step guide: Graphing Linear Equations

Example

The $y$ intercept of a straight line is the value of $y$ when the $x$ -coordinate is zero.

In this real life example, the $y$ intercept represents the starting value or fixed price, with the slope or gradient representing the unit rate or rate of change.

Let’s look at some examples.

A straight line has the equation, in standard form, $4x-3y=18.$ Find the $y$ intercept of the line.

To find the $y$ intercept, substitute $x=0$ into the equation.

4(0)-3y=18

Now solve the equation to find the corresponding $y$ value.

\begin{aligned}-3y&=18 \\\\ y&=-6 \end{aligned}

The $y$ intercept is $-6$ and has coordinate $(0,-6).$

Note: If any function is of the form $y=f(x)+\text{constant},$ the constant is the $y$ intercept. This is because the $y$ -intercept is always when the $x$ -value is $0,$ so when substituting $0$ for $x$ the constant is going to be the $y$ -value.

See examples of different types of polynomials below.

What is the y intercept and how do you find the y-intercept?

Common Core State Standards

How does this relate to high school math?

High School: Functions (HS.IF.C.7a)
Graph linear and quadratic functions and show intercepts, maxima, and minima.

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How to find the y intercept

In order to find the $y$ intercept, you need to:

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.
Solve the equation for $\textbf{y}.$

How to find y intercept examples

Example 1: finding the y intercept of a line in the form y = mx + b

Find the $y$ intercept of the line $y=2x-5.$

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.

y=2(0)-5

2Solve the equation for $\textbf{y}.$

This equation gives $y=-5.$

The $y$ intercept is $-5.$

It has coordinates $(0,-5).$

Note: This is the special case where the equation is in the form $y =$ [function of $x$ ] $+ \; c$ (constant). In this case $c=-5$ so the $y$ intercept is $(0,-5).$

Example 2: finding the y intercept of a line in the form y = mx + b

Find the $y$ intercept of the line $y=\cfrac{1}{2}x+3$

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.

y=\cfrac{1}{2}(0)+3

Solve the equation for $\textbf{y}.$

This equation gives $y=3.$

The $y$ intercept is $3.$

It has coordinates $(0,3).$

Example 3: finding the y intercept of a line in the form y = mx + b

Find the $y$ intercept of the line $y=9x-14.$

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.

y=9(0)-14

Solve the equation for $\textbf{y}.$

This equation gives $y=-14.$

The $y$ intercept is $-14.$

It has coordinates $(0,-14).$

Example 4: finding the y intercept of a line in the form ax + by = c

Find the $y$ intercept of the line $2x+5y=20.$

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.

2(0)+5y=20

Solve the equation for $\textbf{y}.$

\begin{aligned}5y&=20 \\\\ y&=4 \end{aligned}

The $y$ intercept is $4.$

It has coordinates $(0,4).$

Example 5: finding the y intercept of a line in the form ax + by = c

Find the $y$ intercept of the line $3x-4y=24.$

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.

3(0)-4y=24

Solve the equation for $\textbf{y}.$

\begin{aligned}-4y&=24 \\\\ y&=-6 \end{aligned}

The $y$ intercept is $-6.$

It has coordinates $(0,-6).$

Example 6: finding the y intercept of a line in the form ax + by = c

Find the $y$ intercept of the line $7x+9y=36.$

Substitute $\textbf{x} =$ $\bf{0}$ into the equation of the line.

7(0)+9 y=36

Solve the equation for $\textbf{y}.$

\begin{aligned}9y&=36 \\\\ y&=4 \end{aligned}

The $y$ intercept is $4.$

It has coordinates $(0,4).$

Teaching tips for how to find y intercept

While introducing the concept of the $y$ -intercept, use multiple visuals where students can clearly see that the $y$ -intercept is the point where the line crosses the $y$ -axis.

The use of interactive activities, such as online graphing tools or activities that require students to physically graph activities, are just as effective as worksheets with practice problems.

For students that are struggling, break down the process of finding the $y$ -intercept into simple steps. The use of visuals for each step can aid students in understanding the process.

Easy mistakes to make

Substituting the wrong value for zero
A common error is to think that the $y$ intercept is when $y=0.$ It is important to remember that when $y=0,$ the line will be crossing the $x$ -axis and therefore will give the $x$ intercept.

Forgetting to state the coordinates when asked
If an exam question asks for the $y$ intercept or $x$ intercept, just the value where the line crosses the axes is appropriate. However, sometimes the question will ask for the coordinates of the $y$ intercept. In this case the answer must be given in the form $(0,b)$ for the $y$ intercept, or $(-\cfrac{b}{m},0)$ for the coordinates of the $x$ intercept.

Confusing the intercept with the gradient
When the line is in the form $y=mx+b,$ a common error is to confuse the $y$ intercept, $b$ with the gradient or slope of a line, $m.$ The $y$ intercept will have coordinates $(0,b)$ and the $x$ intercept will have coordinates $(-\cfrac{b}{m},0).$

Practice how to find the y intercept questions

(-2,0)

(0,-2)

(3,0)

(0,3)

Substitute $x=0$ to give

\begin{aligned}y&=3(0)-2 \\\\ y&=-2 \end{aligned}

The coordinate of the $y$ -intercept is $(0,-2).$

(0,-6)

(0,2)

(2,0)

(-6,0)

Substitute $x=0$ to give

\begin{aligned}y&=3(0)-6 \\ y&=-6 \end{aligned}

The coordinate of the $y$ -intercept is $(0,-6).$

\cfrac{8}{3}

-3

-8

8

When $x=0,$

\begin{aligned}y&=8-3(0) \\\\ y&=8\end{aligned}

-3

3

-\cfrac{1}{3}

9

When $x=0,$

\begin{aligned}-3y&=9 \\\\ y&=-3 \end{aligned}

2

-2.5

-\cfrac{2}{5}

\cfrac{1}{2}

When $x=0,$

\begin{aligned}5(0)-4y&=10 \\\\ -4y&=10 \\\\ y&=-2.5 \end{aligned}

\cfrac{1}{2}

\cfrac{1}{6}

-2

3

When $x=0,$

\begin{aligned}-2(0)+12y&=6 \\\\ 12y&=6 \\\\ y&=\cfrac{6}{12}=\cfrac{1}{2} \end{aligned}

How to find y intercept FAQs

How do I find the $\textbf{x}$ -intercept of a line?

The process for finding the $x$ -intercept of a line is similar to finding the $y$ -intercept. Instead of substituting $x=0,$ you would substitute $y=0,$ and solve the equation for $x.$

Can you find the $\textbf{y}$ -intercept when the equation is in point-slope form?

Yes, the process for finding the $y$ -intercept in point-slope form is similar to any other linear equation. You will substitute $x=0$ into the equation and then solve for $y.$

How do you find the slope of the line?

To find the slope of the line, you will use the formula: $m=\cfrac{y_{2}-y_{1}}{x_{2}-x_{1}},$ where the slope $(m),$ is found by inserting the $x$ -coordinate and $y$ -coordinate from two points on a given line.

The next lessons are

Rate of change
Systems of equations
Number patterns
Geometry
Angles
Angles in parallel lines

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