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Chapter 2: Problem 93

For the following exercises, find the x- and y-intercepts of each equation $$7 x+2 y=56$$

Short Answer

Expert verified
x-intercept: 8, y-intercept: 28

Step by step solution

01

Finding the x-intercept

The x-intercept occurs when the y-value is zero. To find the x-intercept, substitute \(y = 0\) into the equation \(7x + 2y = 56\). The equation becomes \(7x + 2(0) = 56\), simplifying to \(7x = 56\). Solve for \(x\) by dividing both sides by 7: \(x = \frac{56}{7} = 8\). Therefore, the x-intercept is \(x = 8\).
02

Finding the y-intercept

The y-intercept occurs when the x-value is zero. To find the y-intercept, substitute \(x = 0\) into the equation \(7x + 2y = 56\). The equation becomes \(7(0) + 2y = 56\), simplifying to \(2y = 56\). Solve for \(y\) by dividing both sides by 2: \(y = \frac{56}{2} = 28\). Therefore, the y-intercept is \(y = 28\).

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Key Concepts

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

Linear Equations
Linear equations are equations that form a straight line when plotted on a graph. This is achieved when each term in the equation is either a constant or the product of a constant and a single variable. Commonly, linear equations are expressed in the standard form: \[ax + by = c\] where \(x\) and \(y\) are variables, and \(a\), \(b\), and \(c\) are constants. The equation given in the exercise \(7x + 2y = 56\) is a perfect example of this form. Linear equations are often used to model real-world scenarios and can help in predicting outcomes by examining the relationship between the variables. Understanding linear equations is crucial as it forms the basis for more complex mathematical concepts and applications.
Solving Equations
Solving an equation means finding the value of the variable that makes the equation true. For a linear equation in two variables, like \(7x + 2y = 56\), solving involves finding specific values for \(x\) and \(y\) that satisfy the equation. To solve for the x-intercept:
  • Set \(y = 0\).
  • Substitute into the equation: \(7x + 2(0) = 56\).
  • Solve for \(x\): Divide both sides by 7, resulting in \(x = 8\).
For the y-intercept:
  • Set \(x = 0\).
  • Substitute into the equation: \(7(0) + 2y = 56\).
  • Solve for \(y\): Divide both sides by 2, resulting in \(y = 28\).
These steps show how substituting one variable for zero simplifies finding the intercepts on their respective axes. This method is extremely useful in graphing the line represented by the equation.
Graphing Intercepts
Graphing intercepts involves plotting them on the coordinate plane. The intercepts are where the graph crosses the x-axis and y-axis. These points are key in sketching the line of a linear equation. From our exercise, the x-intercept is \((8, 0)\) and the y-intercept is \((0, 28)\). To graph these intercepts:
  • First, plot the x-intercept \((8, 0)\) on the x-axis.
  • Then, plot the y-intercept \((0, 28)\) on the y-axis.
Draw a straight line crossing both points. This line represents the equation \(7x + 2y = 56\). By understanding intercepts and how to graph them, you can visually interpret the solution of the linear equation. This graphing technique is especially useful to see the relationship between variables and predict trends.

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Most popular questions from this chapter

When hired at a new job selling electronics, you are given two pay options: \(\cdot\) Option A: Base salary of \(\$ 10,000\) a year with a commission of 4\(\%\) of your sales \(\cdot\) Option B: Base salary of \(\$ 20,000\) a year with a commission of 4\(\%\) of your sales How much electronics would you need to sell for option A to produce a larger income?

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years: $$(4,500,2000) ;(4,700,2001) ;(5,200,2003) ;(5,800,2006)$$ Predict when the population will hit 20,000.

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Determine whether the function is increasing or decreasing. $$f(x)=7 x-2$$

Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in Table 2.17. Plot the points, then sketch a line that fits the data. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Predicted } & {6} & {7} & {7} & {8} & {7} & {9} & {10} & {10} \\ \hline \text { Actual } & {6} & {7} & {8} & {8} & {9} & {10} & {10} & {9} \\ \hline\end{array}$$
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