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Chapter 3: Problem 68

Find all intercepts for each line. Some of these lines have only one intercept. $$x+2 y=100$$

Short Answer

Expert verified
The intercepts are (100, 0) and (0, 50).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Starting with the given equation:\[ x + 2y = 100 \]Set \( y = 0 \):\[ x + 2(0) = 100 \]Simplify to obtain \( x = 100 \)Thus, the x-intercept is at the point (100, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Starting again with the given equation:\[ x + 2y = 100 \]Set \( x = 0 \):\[ 0 + 2y = 100 \]Simplify and solve for y:\[ 2y = 100 \]\[ y = 50 \]Therefore, the y-intercept is at the point (0, 50).

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Key Concepts

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

x-intercept
The x-intercept is where a line crosses the x-axis. At this point, the value of y is always zero.
To find the x-intercept of the linear equation \(x + 2y = 100\), we set \(y\) to zero and solve for \(x\).
Starting with the given equation, set \(y = 0\):
\[ x + 2(0) = 100 \]
Simplify to get:
\[ x = 100 \]
This means the x-intercept is at the point (100, 0).
To summarize:
  • The x-intercept occurs where \(y = 0\).
  • Substitute \(y = 0\) into the equation to find \(x\).
  • The x-intercept for this line is at \(x = 100\).
y-intercept
The y-intercept is where a line crosses the y-axis. At this point, the value of x is always zero.
To determine the y-intercept of the linear equation \(x + 2y = 100\), we set \(x\) to zero and solve for \(y\).
Starting with the given equation, set \(x = 0\):
\[ 0 + 2y = 100 \]
Simplify and solve for \(y\):
\[ 2y = 100 \]
\[ y = 50 \]
This means the y-intercept is at the point (0, 50).
Key points to remember:
  • The y-intercept occurs where \(x = 0\).
  • Substitute \(x = 0\) into the equation to find \(y\).
  • The y-intercept for this line is at \(y = 50\).
linear equations
Linear equations describe relationships between two variables with a straight line when graphed. They are typically written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
The general steps to analyze a linear equation such as \(x + 2y = 100\) are:
  • Identify the coefficients of \(x\) and \(y\), and the constant term.
  • To find intercepts, set one variable to zero and solve for the other.
  • Graph the equation using the intercepts to see how the line behaves.
Understanding linear equations is essential as they form the basics of algebra and serve as the foundation for more complex mathematical concepts.
Always remember these key attributes of linear equations:
  • They create straight lines on a graph.
  • Only have powers of one for both variables \(x\) and \(y\).
  • Finding intercepts helps to easily visualize the line's placement on the graph.

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Most popular questions from this chapter

Solve each problem. An office manager is placing an order for note pads at \(\$ 1\) each and binders at \(\$ 2\) each. The total cost of the order must be \(\$ 100 .\) Write an equation for the total cost and graph it. If he orders 30 note pads, then how many binders must he order?

Find all intercepts for each line. Some of these lines have only one intercept. $$9 x+8 y=72$$

Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=\frac{3}{4} x+3\\\ &y=-\frac{4}{3} x+1 \end{aligned} $$

Solve each problem. See Example 9. Carbon dioxide emission. Worldwide emission of carbon dioxide (CO \(_{2}\) ) increased linearly from 14 billion tons in 1970 to 26 billion tons in 2000 (World 91影视 Institute, www.wri.org). a) Express the emission as a linear function of the year in the form \(y=m x+b,\) where \(y\) is in billions of tons and \(x\) is the year. [ Hint: Write the equation of the line through \((1970,14) \text { and }(2000,26) .]\) b) Use the function from part (a) to predict the worldwide emission of \(\mathrm{CO}_{2}\) in 2010 .

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$h(0) \cdot g(0)$$
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