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Chapter 3: Problem 26

Find the intercepts and graph them. $$ 34 x-12 y=-3 $$

Short Answer

Expert verified
X-intercept: \( x = \frac{-3}{34} \), Y-intercept: \( y = \frac{1}{4} \).

Step by step solution

01

Identify the Equation Format

The given equation is in standard form: \( Ax + By = C \). In this case, \( A = 34 \), \( B = -12 \), and \( C = -3 \). Our task is to find both the x-intercept and y-intercept from this equation.
02

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation because the x-intercept is where the graph crosses the x-axis.Setting \( y = 0 \), we get:\[ 34x - 12(0) = -3 \]This simplifies to:\[ 34x = -3 \]Now, solve for \( x \):\[ x = \frac{-3}{34} \]
03

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation because the y-intercept is where the graph crosses the y-axis.Setting \( x = 0 \), we get:\[ 34(0) - 12y = -3 \]This simplifies to:\[ -12y = -3 \]Now, solve for \( y \):\[ y = \frac{-3}{-12} = \frac{1}{4} \]
04

Plot the Intercepts on the Graph

Plot the intercepts on the Cartesian plane:- The x-intercept is \( (\frac{-3}{34}, 0) \), which is very close to the y-axis.- The y-intercept is \( (0, \frac{1}{4}) \).Draw a straight line through these two points. The line represents the graph of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the x-intercept
The x-intercept is an essential concept in algebra and graphing. It is the point where a graph crosses the x-axis. To find the x-intercept, you make the y-value zero in the equation. This is because every point on the x-axis has a y-value of zero.

In our example, the equation given is in the form of \(34x - 12y = -3\). To determine the x-intercept:
  • Substitute 0 for \(y\).
  • The equation simplifies to \(34x = -3\).
  • Solving for \(x\), we find \(x = \frac{-3}{34}\).
This result gives us the exact point on the x-axis where the line crosses, which is very close to the origin due to the small value of \(\frac{-3}{34}\). Understanding this point helps in sketching the graph quickly.
Understanding the y-intercept
The y-intercept is another crucial point in graphing linear equations. It is the location where the graph crosses the y-axis. Finding it requires setting the x-value to zero in the equation because every point on the y-axis has an x-value of zero.

For the given equation, \(34x - 12y = -3\):
  • We substitute 0 for \(x\).
  • The equation simplifies to \(-12y = -3\).
  • Solving for \(y\), we find \(y = \frac{1}{4}\).
This gives the y-intercept which is \((0, \frac{1}{4})\). This point shows where the line meets the y-axis, providing a second vital reference point for drawing the equation's graph.
Graphing the linear equation
Graphing a linear equation like \(34x - 12y = -3\) involves plotting the x-intercept and y-intercept and drawing a line through them. This line represents all the solutions of the equation.

Once you have the intercepts:
  • The x-intercept is \(\left(\frac{-3}{34}, 0\right)\).
  • The y-intercept is \((0, \frac{1}{4})\).
Plot these points on the Cartesian coordinate plane. Then, draw a straight line connecting them. This visual graph represents the relationship described by the equation, showing how x-values and y-values correlate.

Understanding how to find and plot intercepts simplifies graphing, making it easier to visualize and solve linear equations.

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