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Chapter 3: Problem 65

Find at least five ordered pair solutions and graph. $$ 110 x-y=3 $$

Short Answer

Expert verified
Choose values for \(x\), compute \(y = 110x - 3\) for five pairs, and graph them.

Step by step solution

01

Understanding the Equation

We have the equation given as \(110x - y = 3\). This is a linear equation in two variables, \(x\) and \(y\). Our task is to find at least five ordered pairs \((x, y)\) that satisfy this equation. These pairs will be plotted on a graph.
02

Isolate y in the Equation

To find solutions more easily, let's express \(y\) in terms of \(x\). We rearrange the equation: \(y = 110x - 3\). This form makes it easier to calculate \(y\) for different values of \(x\).
03

Calculate Ordered Pairs

Now, choose different values for \(x\) and calculate the corresponding \(y\) values using \(y = 110x - 3\).

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Key Concepts

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

Understanding Ordered Pairs
Ordered pairs are essential in mathematics, especially when working with coordinate systems and graphs. An ordered pair is written in the form \(x, y\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate. These pairs help locate points precisely on a graph.
To find ordered pairs that satisfy an equation like \(110x - y = 3\), you can select different values for \(x\) and then solve for \(y\). Each choice of \(x\) gives a corresponding \(y\), resulting in an ordered pair. For instance, if we choose \(x = 1\), we substitute it into the equation solved for \(y\), \(y = 110 \cdot 1 - 3\), yielding \(y = 107\), and thus the ordered pair \( (1, 107)\).
  • To generate multiple ordered pairs, repeat this process with different \(x\) values.
  • The variety in \(x\) values results in a set of ordered pairs that collectively graph the equation.
Graphing Linear Equations
Graphing linear equations is a visual method to represent solutions of an equation. When dealing with the equation \(110 x - y = 3\), once it is solved for \(y\), you can plot the ordered pairs obtained. Graphing helps to better understand the relationship between the variables.
To graph this equation:
  • Begin by plotting the ordered pairs on the coordinate plane.
  • Each pair \( (x, y)\) determines a point on the graph.
  • Once all points are plotted, draw a straight line through them.
    This line represents all possible solutions to the equation.
Graphing allows you to visually inspect properties of the equation, like slope, intercepts, and consistency of the solutions. This approach provides a comprehensive view beyond numerical values, making the concepts more tangible.
Solving for Y
The process of solving for \(y\) in a linear equation is crucial for understanding its graphical representation. Solving for \(y\) typically involves rearranging terms so that \(y\) stands alone on one side of the equation.
For the equation \(110x - y = 3\), you can solve for \(y\) as follows:
Rearrange the terms to isolate \(y\):\(y = 110x - 3\).
This format, known as the slope-intercept form, \(y = mx + b\), is helpful for identifying:
  • The slope (m), which in our equation is 110.
    This tells you how steep the line is.
  • The y-intercept (b), which is -3.
    This tells you where the line crosses the \(y\)-axis.
By expressing a linear equation in terms of \(y\), you simplify the graphing process and make it easier to understand the linear relationship between \(x\) and \(y\).

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