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Chapter 3: Problem 51

Complete the table of ordered pairs for each linear equation. See Examples 6 through 8 . \(y=-x+2\) \(\begin{array}{|c|c|}\hline x & {y} \\ \hline 0 & {} \\ \hline & {0} \\\ \hline-3 & {} \\ \hline\end{array}\)

Short Answer

Expert verified
The completed table: \((0, 2), (2, 0), (-3, 5)\).

Step by step solution

01

Find y when x = 0

Substitute \(x = 0\) into the equation \(y = -x + 2\). Then \(y = -(0) + 2 = 2\). So the ordered pair is \((0, 2)\).
02

Find x when y = 0

Substitute \(y = 0\) into the equation \(y = -x + 2\) and solve for \(x\). This gives \(0 = -x + 2\). Solving, \(x = 2\). So, the ordered pair is \((2, 0)\).
03

Find y when x = -3

Substitute \(x = -3\) into the equation \(y = -x + 2\). This gives \(y = -(-3) + 2 = 3 + 2 = 5\). So, the ordered pair is \((-3, 5)\).

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Key Concepts

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

Ordered Pairs
Ordered pairs are groups of two numbers used to show a position on a grid. Each pair is written in the form \(x, y\). The first number represents the horizontal position, and the second represents the vertical position.
  • For example, if you have the pair \(3, 4\), it indicates you are 3 units to the right and 4 units up from the origin on a graph.
  • In math, ordered pairs are crucial to plotting points on a coordinate plane and understanding relationships between two variables.
You use ordered pairs when working with linear equations to show specific solutions. In our exercise example, we found specific x-values and y-values from the equation. By substituting these values into one variable at a time, we created ordered pairs like \(0, 2\), \(2, 0\), and \(-3, 5\).
These points can help visualize the line formed by the equation.
Substitution Method
The substitution method is a strategy used to solve equations, especially systems of equations. It involves replacing a variable with an equivalent expression from another equation. This is particularly useful when you have two equations with two variables.
In our scenario, it was employed to find specific values for x or y to build ordered pairs. First, we substituted \(x = 0\) into \(-x + 2\) to find \(y = 2\). Then, substituting \(y = 0\) helped us solve for \(x\).
  • By replacing variables, you aim to simplify the problem, making it easier to find particular solutions.
  • This method is straightforward and effective for quickly finding specific values in equations.
Practicing the substitution method can improve your problem-solving skills and make working with linear equations more manageable.
Solving Equations
Solving equations involves finding the value of variables that make the equation true. Linear equations, like our example \(y = -x + 2\), are first-degree equations (highest power of the variable) and typically involve two variables, x and y.
The process often includes isolating a variable by balancing operations on each side of the equation. Our exercise example showed how to find exact \(x\) or \(y\) values:
  • To find \(y\), set \(x = 0\) and solve: \(-x + 2\) becomes \(-0 + 2\) leading to \(y = 2\).
  • To find \(x\), set \(y = 0\) and solve: \(0 = -x + 2\) leads to \(x = 2\).
Understanding how to solve these equations allows you to determine precise points that fit the line represented by the equation. This foundational skill is vital in algebra, constructions in physics, and other scientific applications.

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Most popular questions from this chapter

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . In January 2007 , there were \(71,000\) registered gasolineelectric hybrid cars in the United States. In \(2004,\) there were only \(29,000\) registered gasoline- electric hybrids. (Source: U.S. Energy Information Administration) a. Write an equation describing the relationship between time and number of registered gasoline-hybrid cars. Use ordered pairs of the form (years past \(2004,\) number of cars). b. Use this equation to predict the number of gasolineelectric hybrids in the year 2010 .

There were approximately \(14,774\) kidney transplants performed in the United States in \(2002 .\) In \(2006,\) the number of kidney transplants performed in the United States rose to 15. \(722 .\) (Source: Organ Procurement and Transplantation Network) a. Write two ordered pairs of the form (year, number of kidney transplants). b. Find the slope of the line between the two points. c. Write a sentence explaining the meaning of the slope as a rate of change.

Write an equation in standard form of the line that contains the point \((-2,4)\) and is a. parallel to the line \(x+3 y=6\) b. perpendicular to the line \(x+3 y=6\)

For Exercises 87 through \(91,\) fill in each blank with "0, "positive," or "negative." For Exercises 92 and \(93,\) fill in each blank with "x" or "y." (__,__) quadrant III

Example $$\text { If } f(x)=x^{2}+2 x+1, \text { find } f(\pi)$$ Solution: $$\begin{aligned} &f(x)=x^{2}+2 x+1\\\ &f(\pi)=\pi^{2}+2 \pi+1 \end{aligned}$$ Given the following functions, find the indicated values. $$f(x)=x^{2}-12$$ a. \(f(12)\) b. \(f(a)\)
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