Find the x x and y y -intercepts of the line 2 x + 3 y = 12 2x+3y=12 .

x − i n t : ( 6 , 0 ) x-int:\left(6,0\right) x − in t : ( 6 , 0 ) y − i n t : ( 0 , 4 ) y-int:\left(0,4\right) y − in t : ( 0 , 4 )

x − i n t : ( 4 , 0 ) x-int:\left(4,0\right) x − in t : ( 4 , 0 ) y − i n t : ( 0 , 6 ) y-int:\left(0,6\right) y − in t : ( 0 , 6 )

x − i n t : ( 2 , 0 ) x-int:\left(2,0\right) x − in t : ( 2 , 0 ) y − i n t : ( 0 , 3 ) y-int:\left(0,3\right) y − in t : ( 0 , 3 )

x − i n t : ( 0 , 2 ) x-int:\left(0,2\right) x − in t : ( 0 , 2 ) y − i n t : ( 3 , 0 ) y-int:\left(3,0\right) y − in t : ( 3 , 0 )

Find the x x and y y -intercepts of the line x 4 − y 3 = 1 \frac{x}{4}-\frac{y}{3}=1 .

x − i n t : ( 4 , 0 ) x-int:\left(4,0\right) x − in t : ( 4 , 0 ) y − i n t : ( 0 , − 3 ) y-int:\left(0,-3\right) y − in t : ( 0 , − 3 )

x − i n t : ( 4 , 0 ) x-int:\left(4,0\right) x − in t : ( 4 , 0 ) y − i n t : ( 0 , 3 ) y-int:\left(0,3\right) y − in t : ( 0 , 3 )

x − i n t : ( − 4 , 0 ) x-int:\left(-4,0\right) x − in t : ( − 4 , 0 ) y − i n t : ( 0 , 3 ) y-int:\left(0,3\right) y − in t : ( 0 , 3 )

x − i n t : ( − 4 , 0 ) x-int:\left(-4,0\right) x − in t : ( − 4 , 0 ) y − i n t : ( 0 , − 3 ) y-int:\left(0,-3\right) y − in t : ( 0 , − 3 )

4. Graphs and Functions

Graph Linear Equations Using Intercepts

4. Graphs and Functions

Graph Linear Equations Using Intercepts: Videos & Practice Problems

Video Lessons

Topic summary

Understanding the x and y intercepts of a line is essential in graphing and analyzing polynomials and linear equations. The x intercept occurs where the line crosses the x-axis, with the y value always zero, while the y intercept occurs where the line crosses the y-axis, with the x value zero. To find these intercepts algebraically, set $y = 0$ to find the x intercept and $x = 0$ to find the y intercept, then solve for the remaining variable. This method applies to equations like $x + 2 y = 8$ , reinforcing key concepts in polynomial terms and standard form.

concept

Identifying x & y Intercepts

Video duration:

Identifying x & y Intercepts Video Summary

Understanding the characteristics of a line on a graph is essential, especially when identifying the x-intercept and y-intercept. These intercepts represent the points where the line crosses the x-axis and y-axis, respectively. The x-intercept occurs where the y-value is zero, so the point can be expressed as (x, 0). Conversely, the y-intercept occurs where the x-value is zero, represented as (0, y).

For example, if a line crosses the x-axis at 4, the x-intercept is (4, 0). If it crosses the y-axis at -2, the y-intercept is (0, -2). These points are crucial for graphing lines and understanding their behavior.

When a graph is not available, the intercepts can still be found algebraically from the equation of the line. To find the x-intercept from an equation, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. This method leverages the fact that at the x-intercept, the line touches the x-axis (where y is zero), and at the y-intercept, it touches the y-axis (where x is zero).

Consider the linear equation $x + 2y = 8$. To find the x-intercept, substitute $y = 0$:

\[x + 2(0) = 8 \implies x = 8\]

This gives the x-intercept as the point $(8, 0)$.

To find the y-intercept, substitute $x = 0$:

\[0 + 2y = 8 \implies 2y = 8 \implies y = \frac{8}{2} = 4\]

This yields the y-intercept as the point $(0, 4)$.

Mastering how to find x and y intercepts both graphically and algebraically enhances your ability to analyze linear equations and their graphs effectively. This foundational skill supports deeper understanding of linear functions and their applications in various mathematical contexts.

example

Identifying x & y Intercepts Example 1

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Identifying x & y Intercepts Example 1 Video Summary

To find the x-intercept and y-intercept of a line on a graph, it is important to understand where the line crosses the coordinate axes. The x-intercept occurs where the line crosses the x-axis, which means the y-coordinate is zero at this point. Conversely, the y-intercept occurs where the line crosses the y-axis, meaning the x-coordinate is zero at this point.

For example, if a line crosses the x-axis at the point where x = 4, then the x-intercept is at the coordinate (4, 0). This indicates that when the value of y is zero, x equals 4. Similarly, if the line crosses the y-axis at the point where y = 3, then the y-intercept is at the coordinate (0, 3). This means that when x is zero, y equals 3.

Identifying these intercepts is crucial for graphing linear equations and understanding their behavior. The intercepts provide key points that help in plotting the line accurately and analyzing its slope and position relative to the axes.

Problem

Find the $x$ and $y$ -intercepts of the line $2 x + 3 y = 12$ .

$x-int:\left(4,0\right)$

$y-int:\left(0,6\right)$

$x-int:\left(2,0\right)$

$y-int:\left(0,3\right)$

$x-int:\left(6,0\right)$

$y-int:\left(0,4\right)$

$x-int:\left(0,2\right)$

$y-int:\left(3,0\right)$

Problem

Find the $x$ and $y$ -intercepts of the line $\frac{x}{4} - \frac{y}{3} = 1$ .

$x-int:\left(4,0\right)$

$y-int:\left(0,3\right)$

$x-int:\left(4,0\right)$

$y-int:\left(0,-3\right)$

$x-int:\left(-4,0\right)$

$y-int:\left(0,3\right)$

$x-int:\left(-4,0\right)$

$y-int:\left(0,-3\right)$

Here’s what students ask on this topic:

To find the x-intercept of a linear equation, you set the y-value to zero and solve for x. This is because the x-intercept is the point where the line crosses the x-axis, and at this point, the y-coordinate is always zero. For example, given the equation $x + 2 y = 8$ , set $y = 0$ . The equation becomes $x + 2 \times 0 = 8$ , which simplifies to $x = 8$ . So, the x-intercept is at the point (8, 0). This method works for any linear equation and is a fundamental step in graphing lines using intercepts.

To find the y-intercept of a linear equation, you set the x-value to zero and solve for y. The y-intercept is the point where the line crosses the y-axis, and at this point, the x-coordinate is always zero. For example, with the equation $x + 2 y = 8$ , substitute $x = 0$ . The equation becomes $0 + 2 y = 8$ . Solving for y, divide both sides by 2: $y = \frac{8}{2} = 4$ . Thus, the y-intercept is at (0, 4). This approach is essential for graphing linear equations and understanding their behavior on the coordinate plane.

Yes, you can find the x and y intercepts without a graph by using algebraic methods. To find the x-intercept, set $y = 0$ in the equation and solve for $x$ . To find the y-intercept, set $x = 0$ and solve for $y$ . For example, given the equation $x + 2 y = 8$ , setting $y = 0$ gives $x = 8$ , so the x-intercept is (8, 0). Setting $x = 0$ gives $2 y = 8$ , so $y = 4$ , and the y-intercept is (0, 4). This method is useful when a graph is not available.

The y-value is zero when finding the x-intercept because the x-intercept is the point where the line crosses the x-axis. On the coordinate plane, the x-axis is the horizontal axis where the y-coordinate is always zero. Therefore, any point on the x-axis has a y-value of zero. When you set $y = 0$ in the equation of a line, you are identifying the x-coordinate where the line intersects the x-axis. This is why the y-value is zero for the x-intercept, making it easier to solve for the corresponding x-value algebraically.

To graph a linear equation using intercepts, first find the x-intercept by setting $y = 0$ and solving for $x$ . Then find the y-intercept by setting $x = 0$ and solving for $y$ . Plot these two points on the coordinate plane. For example, for the equation $x + 2 y = 8$ , the x-intercept is (8, 0) and the y-intercept is (0, 4). After plotting these points, draw a straight line through them. This line represents the graph of the equation. Using intercepts is a quick and effective way to graph linear equations without needing to calculate multiple points.