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

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

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

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

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

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)

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

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

4. Graphing Linear Equations in Two Variables

Graph Linear Equations Using Intercepts

4. Graphing Linear Equations in Two Variables

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. To find these intercepts algebraically, set $y = 0$ to find the x-intercept and $x = 0$ for the y-intercept, then solve for the remaining variable. This method applies to polynomials and linear equations, reinforcing key concepts like terms, coefficients, and standard form.

concept

Identifying x & y Intercepts

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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 $x = 4$, the x-intercept is the point $(4, 0)$. Similarly, if the line crosses the y-axis at $y = -2$, 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 $y$. For instance, given the linear equation:

\[x + 2y = 8,\]

finding the x-intercept involves substituting $y = 0$:

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

This gives the x-intercept as $(8, 0)$. To find the y-intercept, substitute $x = 0$:

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

resulting in the y-intercept $(0, 4)$.

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.

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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 is where the line crosses the y-axis, so the x-coordinate is zero there.

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 y = 3, the y-intercept is at (0, 3), meaning when x is zero, y equals 3.

Identifying these intercepts is essential 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 $2 x + 3 y = 12$ .

$x minus i n t colon open paren 4 comma 0 close paren$

$y minus i n t colon open paren 0 comma 6 close paren$

$x minus i n t colon open paren 2 comma 0 close paren$

$y minus i n t colon open paren 0 comma 3 close paren$

$x minus i n t colon open paren 6 comma 0 close paren$

$y minus i n t colon open paren 0 comma 4 close paren$

$x minus i n t colon open paren 0 comma 2 close paren$

$y minus i n t colon open paren 3 comma 0 close paren$

Problem

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

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

$y minus i n t colon open paren 0 comma 3 close paren$

$x minus i n t colon open paren 4 comma 0 close paren$

$y - i n t : (0, - 3)$

$x - i n t : (- 4, 0)$

$y minus i n t colon open paren 0 comma 3 close paren$

$x - i n t : (- 4, 0)$

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

Here’s what students ask on this topic:

To find the x-intercept of a linear equation, set $y = 0$ and solve for $x$ . This gives the point where the line crosses the x-axis, so the y-value is zero. For the y-intercept, set $x = 0$ and solve for $y$ . This gives the point where the line crosses the y-axis, so the x-value is zero. For example, given the equation $x + 2 y = 8$ , to find the x-intercept, set $y = 0$ : $x + 2 \times 0 = 8$ which simplifies to $x = 8$ . The x-intercept is (8, 0). For the y-intercept, set $x = 0$ : $0 + 2 y = 8$ , solving gives $y = 4$ . The y-intercept is (0, 4).

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 x-axis, all points have a y-coordinate of zero by definition. This means the line touches or crosses the x-axis at some point where $y = 0$ . By substituting $y = 0$ into the equation of the line, you can solve for the corresponding x-value, which gives the exact location of the x-intercept. This concept is fundamental in graphing linear equations and helps in understanding the behavior of the line relative to the coordinate axes.

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

The x and y intercepts are significant because they provide key points where the line crosses the coordinate axes, making it easier to graph the line 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 intercepts allows you to plot two points on the graph, which is sufficient to draw the entire line. Additionally, intercepts help in understanding the behavior and position of the line relative to the axes, which is essential in analyzing linear relationships in algebra and real-world applications.

To find the x-intercept, set $y = 0$ in the equation $3 x - 4 y = 12$ . This gives $3 x - 4 \times 0 = 12$ , simplifying to $3 x = 12$ . Dividing both sides by 3, $x = 4$ . So, the x-intercept is (4, 0). To find the y-intercept, set $x = 0$ : $3 \times 0 - 4 y = 12$ , which simplifies to $- 4 y = 12$ . Dividing both sides by -4, $y = - 3$ . So, the y-intercept is (0, -3).