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. Graphing

Graph Linear Equations Using Intercepts

4. Graphing

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 linear equations. The x-intercept occurs where the line crosses the x-axis, with the y-value always zero, while the y-intercept is where the line crosses the y-axis, with the x-value zero. These intercepts can be found graphically or algebraically by setting $y = 0$ to find the x-intercept and $x = 0$ for the y-intercept. This method applies to polynomials and linear expressions, aiding in understanding terms, coefficients, 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 written 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 y-coordinate is zero, and at the y-intercept, the x-coordinate is zero.

Consider the linear equation:

\[x + 2y = 8\]

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

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

Thus, the x-intercept is (8, 0).

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

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

Therefore, the y-intercept is (0, 4).

Mastering how to find x and y intercepts both graphically and algebraically enhances your ability to analyze linear equations and their graphs. This foundational skill supports deeper understanding of linear relationships and prepares you for more advanced topics in algebra and coordinate geometry.

example

Identifying x & y Intercepts Example 1

Video duration:

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 at this point is zero. For example, if the line crosses the x-axis at an x-value of 4, the x-intercept is the point (4, 0).

Similarly, the y-intercept is the point where the line crosses the y-axis, meaning the x-coordinate at this point is zero. If the line crosses the y-axis at a y-value of 3, the y-intercept is the point (0, 3).

Identifying these intercepts is essential for graphing linear equations and understanding their behavior. The intercepts provide key information about the line’s position relative to the axes and can be used to write the equation of the line in slope-intercept form, which is given by the formula:

\[y = mx + b\]

where m is the slope of the line and b is the y-intercept. Knowing the intercepts allows you to plot the line accurately and analyze its properties.

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, if you have the equation $x + 2 y = 8$ , you substitute $y = 0$ to get $x + 2 \times 0 = 8$ , which simplifies to $x = 8$ . So, the x-intercept is at the point (8, 0). This method works for any linear equation and helps you graph the line accurately.

The y-intercept is the point where a line crosses the y-axis, and at this point, the x-value is always zero. To find the y-intercept from an equation, you set $x = 0$ and solve for y. For example, given the equation $x + 2 y = 8$ , substitute $x = 0$ to get $0 + 2 y = 8$ . 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 and understanding linear equations.

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

Intercepts are important in graphing linear equations because they provide key points where the line crosses the axes, making it easier to sketch the graph accurately. The x-intercept shows where the line crosses the x-axis (y=0), and the y-intercept shows where it crosses the y-axis (x=0). Knowing these points helps you plot the line quickly without needing many points. Additionally, intercepts give insight into the behavior of the line and are often used in solving real-world problems involving linear relationships. Understanding intercepts also aids in converting equations into different forms, such as slope-intercept form.

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. Since a line is determined by two points, draw a straight line through these intercepts. For example, for the equation $x + 2 y = 8$ , the x-intercept is (8, 0) and the y-intercept is (0, 4). Plot these points and connect them with a straight line to graph the equation accurately.