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Intercepts of lines review (x-intercepts and y-intercepts)

The x-intercept is where a line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. Thinking about intercepts helps us graph linear equations.

What are intercepts?

The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis.
A coordinate plane. The x- and y-axes each scale by one. The graph of the line is labeled y equals one-half x minus three. The y-intercept is labeled at the point zero, negative three. The x-intercept is labeled the point six, zero.
Want a deeper introduction to intercepts? Check out this video.

Example: Intercepts from a graph

Looking at the graph, we can find the intercepts.
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, four and five, zero.
The line crosses the axes at two points:
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, four and five, zero. Both of these points are plotted.
The point on the x-axis is left parenthesis, 5, comma, 0, right parenthesis. We call this the x-intercept.
The point on the y-axis is left parenthesis, 0, comma, 4, right parenthesis. We call this the y-intercept.
Want to learn more about finding intercepts from graphs? Check out this video.

Example: Intercepts from a table

We're given a table of values and told that the relationship between x and y is linear.
xy
1minus, 9
3minus, 6
5minus, 3
Then we're asked to find the intercepts of the corresponding graph.
The key is realizing that the x-intercept is the point where y, equals, 0, and the y-intercept is where x, equals, 0.
A table of values. The left column is labeled x, and the right column is labeled y. When x is negative one, y is negative twelve. When x is one, y is negative nine. When x is three, y is negative six. When x is five, y is negative three. When x is seven, y is zero. Between every x-value there is a plus two which highlights the change of the x-values. Between every y-value there is a plus three which highlights the change of the y-values.
The point left parenthesis, 7, comma, 0, right parenthesis is our x-intercept because when y, equals, 0, we're on the x-axis.
To find the y-intercept, we need to "zoom in" on the table to find where x, equals, 0.
A table of values. The left column is labeled x, and the right column is labeled y. When x is negative one, y is negative twelve. When x is zero, y is negative ten point five. When x is one, y is negative nine. Between every x-value there is a plus one which highlights the change of the x-values. Between every y-value there is a plus one point five which highlights the change of the y-values.
The point left parenthesis, 0, comma, minus, 10, point, 5, right parenthesis is our y-intercept.
Want to learn more about finding intercepts from tables? Check out this video.

Example: Intercepts from an equation

We're asked to determine the intercepts of the graph described by the following linear equation:
3, x, plus, 2, y, equals, 5
To find the y-intercept, let's substitute start color #6495ed, x, end color #6495ed, equals, start color #6495ed, 0, end color #6495ed into the equation and solve for y:
30+2y=52y=5y=52\begin{aligned}3\cdot\blue{0}+2y&=5\\ 2y&=5\\ y&=\dfrac{5}{2}\end{aligned}
So the y-intercept is left parenthesis, 0, comma, start fraction, 5, divided by, 2, end fraction, right parenthesis.
To find the x-intercept, let's substitute start color #ff00af, y, end color #ff00af, equals, start color #ff00af, 0, end color #ff00af into the equation and solve for x:
3x+20=53x=5x=53\begin{aligned}3x+2\cdot\pink{0}&=5\\ 3x&=5\\ x&=\dfrac{5}{3}\end{aligned}
So the x-intercept is left parenthesis, start fraction, 5, divided by, 3, end fraction, comma, 0, right parenthesis.
Want to learn more about finding intercepts from equations? Check out this video.

Practice

Problem 1
Determine the intercepts of the line graphed below.
x-intercept:
left parenthesis
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0, point, 75
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
comma
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0, point, 75
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
right parenthesis
y-intercept:
left parenthesis
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0, point, 75
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
comma
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0, point, 75
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
right parenthesis
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points negative seven, zero and zero, two.

Want more practice? Check out these exercises:

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