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Magazine Chapter 3: Problem 43
Use the intercept method to graph each equation. $$ 3 x-4 y=11 $$
Short Answer
Expert verified
The graph passes through \((\frac{11}{3}, 0)\) and \((0, -\frac{11}{4})\).
Step by step solution
01
Identify the Intercepts
To find the x-intercept, set \( y = 0 \) and solve the equation \( 3x - 4(0) = 11 \). To find the y-intercept, set \( x = 0 \) and solve the equation \( 3(0) - 4y = 11 \).
02
Calculate the X-intercept
For the x-intercept calculation: \( 3x = 11 \). Dividing both sides by 3 gives \( x = \frac{11}{3} \). The x-intercept is \( (\frac{11}{3}, 0) \).
03
Calculate the Y-intercept
For the y-intercept calculation: \( -4y = 11 \). Dividing both sides by -4 gives \( y = -\frac{11}{4} \). The y-intercept is \( (0, -\frac{11}{4}) \).
04
Plot the Intercepts on the Graph
Using the intercepts \((\frac{11}{3}, 0)\) and \((0, -\frac{11}{4})\), plot these points on a coordinate plane.
05
Draw the Line
Connect the two intercept points with a straight line to represent the equation \( 3x - 4y = 11 \). The line should extend across the graph in both directions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intercept Method
The intercept method is a straightforward way to graph linear equations. It's all about finding the points where the line crosses the x-axis and y-axis. These points are known as the x-intercept and y-intercept. Once you have these two points, graphing the equation becomes a simplified task. You just plot the intercepts on a coordinate plane and connect them with a line.
- Identify Intercepts: Begin by finding the values where the line crosses the axes.
- X-intercept: Set \( y = 0\) and solve for \( x \).
- Y-intercept: Set \( x = 0\) and solve for \( y \).
- Connect the Points: Plot these intercepts on the graph and draw a line through them.
X-Intercept
The x-intercept is a point on the graph where your line crosses the x-axis. At this point, the value of \( y \) is always zero. By setting \( y = 0 \) in your equation, you solve for \( x \) to find the x-intercept.
For example, in the equation \( 3x - 4y = 11 \), you substitute \( y = 0 \), resulting in:
\[ 3x - 4(0) = 11 \]
This simplifies to \( 3x = 11 \). Solving gives \( x = \frac{11}{3} \). Therefore, the x-intercept is \( (\frac{11}{3}, 0) \).
For example, in the equation \( 3x - 4y = 11 \), you substitute \( y = 0 \), resulting in:
\[ 3x - 4(0) = 11 \]
This simplifies to \( 3x = 11 \). Solving gives \( x = \frac{11}{3} \). Therefore, the x-intercept is \( (\frac{11}{3}, 0) \).
- Find the x-value when \( y = 0 \).
- This gives the point where the line touches the x-axis.
Y-Intercept
The y-intercept is another key point on a graph where your line crosses the y-axis. Here, the value of \( x \) is always zero. By setting \( x = 0 \) in the equation, you solve for \( y \) to find the y-intercept.
Consider the same equation \( 3x - 4y = 11 \). Substitute \( x = 0 \), which gives:
\[ 3(0) - 4y = 11 \]
This simplifies to \(-4y = 11 \). Solving for \( y \) gives \( y = -\frac{11}{4} \). Therefore, the y-intercept is \( (0, -\frac{11}{4}) \).
Consider the same equation \( 3x - 4y = 11 \). Substitute \( x = 0 \), which gives:
\[ 3(0) - 4y = 11 \]
This simplifies to \(-4y = 11 \). Solving for \( y \) gives \( y = -\frac{11}{4} \). Therefore, the y-intercept is \( (0, -\frac{11}{4}) \).
- Find the y-value when \( x = 0 \).
- This point is where the line touches the y-axis.
Coordinate Plane
The coordinate plane is a two-dimensional surface where you can plot points defined by pairs of numbers known as coordinates. Each point on the plane has an \(x\ ext{-coordinate}\) and a \(y\ ext{-coordinate}\). The horizontal axis is the x-axis, and the vertical axis is the y-axis. Together, they form a grid that allows you to easily plot and visualize equations.
When graphing using the intercept method, knowing how to use the coordinate plane is crucial. It helps you:
When graphing using the intercept method, knowing how to use the coordinate plane is crucial. It helps you:
- Plot Intercept Points: With the x-intercept and y-intercept found, you can use these coordinates to locate the points on the graph.
- Draw the Line: The line can then be drawn through these plotted points, illustrating the equation's relationship.
