Chapter 5.6, Problem 42E

To determine

To find: the linear inequality in two variable that has following properties.

Answer to Problem 42E

The linear equation is two variables is $y=-x+2$ .

Explanation of Solution

Given:

  $\left(0,0\right),\left(0,-1\right),\left(0,1\right)$ are not the solutions.

  $\left(1,1\right),\left(3,-1\right),\left(-1,3\right)$ are solutions.

Concept used:

The slope of the straight-line equation is denoted by $m$ and equation is in the form of:

  $y=mx+c$ .

Here $m$ is slope and $c$ is $y$ -intercept.

Calculation:

The objective is to find the linear inequation of two variables from the following properties.

  $\left(0,0\right),\left(0,-1\right),\left(0,1\right)$ are not the solutions.

  $\left(1,1\right),\left(3,-1\right),\left(-1,3\right)$ are solutions.

As $\left(1,1\right),\left(3,-1\right),\left(-1,3\right)$ are the solutions following are the steps to write an inequality.

Step 1:

Write the slope from $\left(1,1\right),\left(3,-1\right)$ .

Slope:

  $\begin{array}{l}\left(m\right)=\frac{\text{change in }y}{\text{change in }x}.\\ m=\frac{-1-1}{3-1}=-1.\end{array}$

For y-intercept $\left(b\right)$ :

Equation $y=mx+b$ take a point which is solution $\left(1,1\right)$ :

  $\begin{array}{l}y=mx+b.\\ 1=-1\left(1\right)+b.\\ b=2.\end{array}$

Linear equation is:

  $\begin{array}{l}y=-1\left(x\right)+2.\\ y=-x+2.\end{array}$

Step 2: As $\left(0,0\right),\left(0,-1\right),\left(0,1\right)$ are not the solutions. So, the portion of the graph is shaded which does not contains $\left(0,0\right)$ .

Step $3$ : as the value of $y$ is always greater than or equal to $x$ as clear from the equation. So, the inequality is

The graph of the inequality equation:

  

Hence, the linear equation is two variables is $y=-x+2$ .