Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 7 - Algebra: Graphs, Functions, and Linear Systems - 7.3 Systems of Linear Equations in Two Variables - Exercise Set 7.3 - Page 444: 10

Answer

See below:

Work Step by Step

The provided system of equations is: \[\begin{align} & y=x+1 \\ & y=3x-1 \end{align}\] To solve it by graph method, plot the two lines in rectangular coordinate system. Use the following steps: Step 1: First graph the line \[y=x+1\]. Find x-intercept, set \[y=0\] in above equation. \[\begin{align} & y=x+\grave{\ }1 \\ & 0=x+1 \\ & x=-1 \end{align}\] Find y-intercept, set \[x=0\]in above equation. \[\begin{align} & y=x+1 \\ & y=0+1 \\ & y=1 \end{align}\] Plot the ordered pairs \[\left( -1,0 \right)\text{ and }\left( 0,1 \right)\]. Step2: Draw a line passes through \[\left( -1,0 \right)\]and \[\left( 0,1 \right)\]. Step3: Now graph the line \[y=3x-1\]. Find x-intercept, set \[y=0\] in above equation. \[\begin{align} & y=3x-1 \\ & 0=3x-1 \\ & 3x=1 \\ & x=\frac{1}{3} \end{align}\] Find y-intercept, set \[x=0\]in above equation. \[\begin{align} & y=3x-1 \\ & y=3\cdot 0-1 \\ & y=-1 \end{align}\] Plot the ordered pairs \[\left( \frac{1}{3},0 \right)\text{ and }\left( 0,-1 \right)\]. Step4: Draw a line passes through \[\left( \frac{1}{3},0 \right)\]and \[\left( 0,-1 \right)\]. Step5: From the graph, point of intersection is \[\left( 1,2 \right)\]. To ensure that the graph is accurate, check the point of intersection\[\left( 1,2 \right)\]in both equations. \[\begin{align} & y=x+1 \\ & 2=1+1 \\ & 2=2 \end{align}\] \[\begin{align} & y=3x-1 \\ & 2=3\cdot 1-1 \\ & 2=3-1 \\ & 2=2 \end{align}\] Coordinates of the point of intersection \[\left( 1,2 \right)\]to satisfy both the equations. Hence, the graph of the system of equations is correct.
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