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Chapter 5: Problem 47

Find the \(x\) - and \(y\) -intercepts of the line with the given equation. Sketch the line using the intercepts. A calculator can be used to check the graph. $$12 x+y=30$$

Short Answer

Expert verified
The x-intercept is \(\frac{5}{2}\) and the y-intercept is 30.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the value of \(y\) to 0 in the equation and solve for \(x\):\[12x + 0 = 30\]\[12x = 30\]Divide both sides by 12 to solve for \(x\):\[x = \frac{30}{12} = \frac{5}{2}\]So, the x-intercept is at \(\left(\frac{5}{2}, 0\right)\).
02

Find the y-intercept

To find the y-intercept, set the value of \(x\) to 0 in the equation and solve for \(y\):\[12(0) + y = 30\]\[y = 30\]So, the y-intercept is at \((0, 30)\).
03

Sketch the line using intercepts

Plot the x-intercept \(\left(\frac{5}{2}, 0\right)\) on the x-axis and the y-intercept \((0, 30)\) on the y-axis on a coordinate grid. Draw a straight line through these two points to represent the equation \(12x + y = 30\).
04

Verify the graph with a calculator

Using a graphing calculator, input the equation \(y = -12x + 30\) (rearrange to solve for \(y\)) to check that the graph passes through the points \(\left(\frac{5}{2}, 0\right)\) and \((0, 30)\), verifying the accuracy of our sketch.

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Key Concepts

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

Intercepts
The intercepts of a line are the points where the line crosses the x-axis and y-axis. Determining the intercepts is a foundational skill in understanding linear equations. Here's how you find these important points:
  • The x-intercept occurs where the line crosses the x-axis. At this point, the value of y is zero. To find it, set \(y = 0\) in the equation and solve for \(x\). For the equation \(12x + y = 30\), setting \(y = 0\) gives \(12x = 30\). Solving, we find the x-intercept at \(x = \frac{5}{2}\), or \((\frac{5}{2}, 0)\).
  • The y-intercept occurs where the line crosses the y-axis. At this point, the value of x is zero. To find it, set \(x = 0\) and solve for \(y\). For the same equation, \(12(0) + y = 30\), results in \(y = 30\). Thus, the y-intercept is \((0, 30)\).
Understanding intercepts allows you to quickly sketch linear equations and solve real-world problems. They are essential for graphing lines on a coordinate plane.
Graphing Lines
To draw the line represented by a linear equation, you can start by identifying the intercepts. In our case, the line with the equation \(12x + y = 30\) can be sketched through these simple steps:Firstly, plot the intercepts:
  • Mark the x-intercept \((\frac{5}{2}, 0)\) on the x-axis.
  • Mark the y-intercept \((0, 30)\) on the y-axis.
These two points provide a clear path for visualizing the line.Secondly, connect the intercepts:
  • Draw a straight line through both intercepts. This line is the graphical representation of the equation.
  • Extend the line in both directions beyond the intercepts, maintaining its straight nature, to represent the line across the coordinate plane.
Remember to check that the plotted line accurately represents the equation by substituting the coordinates of other points from the line into the original equation.
Coordinate Plane Visualization
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. Here's a simple guide to visualizing linear equations on it:Understanding the layout:
  • The horizontal line is the x-axis and the vertical line is the y-axis. Their intersection is the origin, \((0, 0)\).
  • Points are represented by ordered pairs \((x, y)\), where \(x\) is the position along the horizontal axis and \(y\) is the position along the vertical axis.
Using the coordinate plane effectively:
  • Plotting intercepts helps anchor the line. The x-intercept and y-intercept show exactly where the line will touch the axes.
  • After plotting, drawing the line gives a complete visual of the linear equation, making abstract numbers tangible and easier to analyze.
  • Checking the line with a graphing calculator reinforces understanding, as many graph calculative tools allow visual overlays on the coordinate plane.
Visualizing equations on the coordinate plane is vital for mathematical literacy, as it allows you to see relationships between variables clearly.

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Most popular questions from this chapter

Set up systems of equations and solve by any appropriate method. All numbers are accurate to at least two significant digits. A person invested a total of \(\$ 20,900\) into two bonds, one with an annual interest rate of \(6.00 \%\) and the other with an annual interest rate of \(5.00 \%\) per year. If the total annual interest from the bonds is \(\$ 1170,\) how much is invested in each bond?

Solve the given systems of equations algebraically. The numbers are approximate. $$\begin{aligned} &32 t+24 u+63 v=32\\\ &19 v-31 u+42 t=132\\\ &48 t+12 u+11 v=0 \end{aligned}$$

Solve the given systems of equations by determinants. $$\begin{aligned} &90 x-110 y=40\\\ &30 x-15 y=25 \end{aligned}$$

Set up appropriate systems of two linear equations in two unknowns and then solve the systems by determinants. All numbers are accurate to at least two significant digits. Two types of electromechanical carburetors are being assembled and tested. Each of the first type requires 15 min of assembly time and 2 min of testing time. Each of the second type requires \(12 \mathrm{min}\) of assembly time and 3 min of testing time. If 222 min of assembly time and 45 min of testing time are available, how many of each type can be assembled and tested, if all the time is used?

Solve the given systems of equations by determinants. $$\begin{aligned} &2 x-3 y=4\\\ &2 x+y=-4 \end{aligned}$$
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