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Chapter 6: Problem 148

Graph using intercepts: \(5 x-2 y=10\) (Section \(3.2,\) Example 4 )

Short Answer

Expert verified
The x-intercept of the given equation is 2 and the y-intercept is -5. These are plotted as points on a graph and a line is drawn to represent the equation.

Step by step solution

01

Finding the X-Intercept

First, the x-intercept is determined by setting \(y = 0\) in the equation and then solving for \(x\). Let's do this:From \(5x - 2*0 = 10\), we get \(x = 2\). Thus, the x-intercept is 2.
02

Finding the Y-Intercept

Next, the y-intercept is determined by setting \(x = 0\) in the equation and then solving for \(y\). Let's do this:From \(5*0 - 2y = 10\), we get \(y = -5\). Therefore, the y-intercept is -5.
03

Graphing The Equation Using The Intercepts

With the x-intercept being (2,0) and the y-intercept being (0,-5), these two points are then plotted on the graph. After the points are plotted, a straight line running through them is drawn to complete the graph of the equation \(5x - 2y = 10\).

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Key Concepts

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

X-Intercept Calculation
Understanding the x-intercept calculation of a graph is essential for illustrating where a line crosses the x-axis. This point indicates where the value of y is zero. The textbook exercise demonstrates this concept with the equation 5x - 2y = 10. To find the x-intercept, you set y to zero and solve for x:

For the given equation, setting y = 0 gives us 5x = 10. Dividing both sides by 5 leaves us with x = 2. Therefore, the x-intercept is the point (2, 0). Placing this on a graph will show where the line intersects the x-axis.
Y-Intercept Calculation
The y-intercept calculation reveals the point where a line crosses the y-axis, corresponding to an x value of zero. For the exercise equation, setting x = 0 yields the simplified form -2y = 10. To find y, we divide both sides by -2, resulting in y = -5. This positions the y-intercept at (0, -5). By identifying this point, learners can quickly determine how the line behaves at the outset along the y-axis when graphed.
Linear Equations
The example provided teaches about linear equations, which are mathematical expressions that describe straight lines on a graph. The general form of a linear equation in two variables is ax + by = c, where a, b, and c are constants, and x and y are variables.In our exercise, the linear equation is 5x - 2y = 10. One can easily identify its slope and intercepts to graph it accurately. Students should remember that linear equations will always result in straight lines, and it's the slope and intercept that define its specific characteristics and direction on the coordinate plane.
Graphing Linear Equations
When graphing linear equations such as the one in this exercise, we rely on the intercepts for a reliable method to draw the line. After calculating the x-intercept (2, 0) and y-intercept (0, -5), you plot these two points on the graph. Then, draw a straight line through these points to represent the linear equation 5x - 2y = 10.This visual representation is pivotal since it allows students to understand the behavior of the equation beyond just the algebraic form. By connecting the intercepts with a ruler, you'll ensure the line is accurate and extends both directions across the graph. Remember, any two points are sufficient to graph a straight line, but the intercepts are often the easiest to find and use.

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Most popular questions from this chapter

If \((x+2)(x-4)=0\) indicates that \(x+2=0\) or \(x-4=0,\) explain why \((x+2)(x-4)=6\) does not mean \(x+2=6\) or \(x-4=6 .\) Could we solve the equation using \(x+2=3\) and \(x-4=2\) because \(3 \cdot 2=6 ?\)

In Exercises \(124-127,\) factor each polynomial. $$x^{2}-y^{2}+3 x+3 y$$

Great white sharks have triangular teeth with a height that is 1 centimeter longer than the base. If the area of one tooth is 15 square centimeters, find its base and height.

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &x^{4}-16=\left(x^{2}+4\right)(x+2)(x-2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

In Exercises \(124-127,\) factor each polynomial. $$x^{2 n}-25 y^{2 n}$$
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