Section 1
Solving Linear Systems by Graphing
Explain what it means to solve a system of linear equations.
Explain how to use the graph at the right to solve the system of linear equations.$$\begin{array}{l}y=-x+2 \\y=x+2\end{array}$$
Graph the linear system below. Then decide if the ordered pair is a solution of the system.$$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$$$(-4,-2)$$
Graph the linear system below. Then decide if the ordered pair is a solution of the system.$$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$$$(4,-2)$$
Graph the linear system below. Then decide if the ordered pair is a solution of the system.$$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$$$(-4,2)$$
Graph the linear system below. Then decide if the ordered pair is a solution of the system.$$ \begin{array}{l} -x+y=-2 \\ 2 x+y=10 \end{array} $$$$(4,2)$$
Confirm your answer for Exercise 3 algebraically.
Use the graph-and-check method to solve the system of linear equations.$$\begin{aligned}&y=2 x-1\\&y=x+1\end{aligned}$$
Use the graph-and-check method to solve the system of linear equations.$$\begin{aligned}&y=-2 x+3\\&y=x-3\end{aligned}$$
Use the graph-and-check method to solve the system of linear equations.$$\begin{aligned}&y=\frac{1}{2} x+2\\&y=-x+5\end{aligned}$$
Decide whether the ordered pair is a solution of the system of linear equations.$$\begin{array}{ll}3 x-2 y=11 & (5,2) \\-x+6 y=7\end{array}$$
Decide whether the ordered pair is a solution of the system of linear equations.$$\begin{array}{ll}6 x-3 y=-15 & (-2,1) \\2 x+y=-3 &\end{array}$$
Decide whether the ordered pair is a solution of the system of linear equations.$$\begin{array}{ll}x+3 y=15 & (3,-6) \\4 x+y=6\end{array}$$
Decide whether the ordered pair is a solution of the system of linear equations.$$\begin{array}{ll}-5 x+y=19 & (-4,-1) \\x-7 y=3\end{array}$$
Decide whether the ordered pair is a solution of the system of linear equations.$$\begin{array}{ll}-15 x+7 y=1 & (3,5) \\3 x-y=1\end{array}$$
Decide whether the ordered pair is a solution of the system of linear equations.$$\begin{array}{ll}-2 x+y=-11 & \\-x-9 y=-15 & (6,1)\end{array}$$
Use the graph to solve the linear system. Check your solution algebraically.(Graph can't copy)$$\begin{aligned}&-x+2 y=6\\&x+4 y=24\end{aligned}$$
Use the graph to solve the linear system. Check your solution algebraically.(Graph can't copy)$$\begin{aligned}&2 x-y=-2\\&4 x-y=-6\end{aligned}$$
Use the graph to solve the linear system. Check your solution algebraically.(Graph can't copy)$$\begin{aligned}&x+y=3\\&-2 x+y=-6\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&y=-x+3\\&y=x+1\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}y &=-6 \\x &=6\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&y=2 x-4\\&y=-\frac{1}{2} x+1\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&2 x-3 y=9\\&x=-3\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&5 x+4 y=16\\&y=-16\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&x-y=1\\&5 x-4 y=0\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&3 x+6 y=15\\&-2 x+3 y=-3\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&7 y=-14 x+42\\&7 y=14 x+14\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&0.5 x+0.6 y=5.4\\&-x+y=9\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&15 x-10 y=-80\\&6 x+8 y=-80\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&-3 x+y=10\\&7 x+y=20\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&x-8 y=-40\\&-5 x+8 y=8\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&\frac{1}{5} x+\frac{3}{5} y=\frac{12}{5}\\&-\frac{1}{5} x+\frac{3}{5} y=\frac{6}{5}\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&\frac{3}{4} x-\frac{1}{4} y=-\frac{1}{2}\\&\frac{1}{4} x-\frac{3}{4} y=\frac{3}{2}\end{aligned}$$
Graph and check to solve the linear system.$$\begin{aligned}&2.8 x+1.4 y=1.4\\&0.7 x-0.7 y=1.4\end{aligned}$$
Car model X costs $\$ 22,000$ to purchase and an average of $\$ .12$ per mile to maintain. Car model $Y$ costs $\$ 24,500$ to purchase and an average of $\$ .10$ per mile to maintain. Use the graph to determine how many miles must be driven for the total cost of the two models to be the same.
A fitness club offers two water aerobics classes. There are currently 40 people regularly going to the morning class, and attendance is increasing at a rate of 2 people per month. There are currently 22 people regularly going to the evening class, and attendance is increasing at a rate of 8 people per month. Predict when the number of people in each class will be the same.
Use the table below, which gives the percents of people in the contiguous United States living within50 miles of a coastal shoreline and those living further inland.$$\begin{array}{|l|c|c|}\hline \text { Qummoninititicustics } & \text { 1940 } & \text { 1997 } \\\hline \begin{array}{l}\text { Living within 50 miles } \\\text { of a coastal shoreline }\end{array} & 46 \% & 53 \% \\\hline \text { Living farther inland } & 54 \% & 47 \% \\\hline\end{array}$$For each location, write a linear model to represent the percent at time $t$ where $t$ represents the number of years since 1940
Use the table below, which gives the percents of people in the contiguous United States living within50 miles of a coastal shoreline and those living further inland.$$\begin{array}{|l|c|c|}\hline \text { Qummoninititicustics } & \text { 1940 } & \text { 1997 } \\\hline \begin{array}{l}\text { Living within 50 miles } \\\text { of a coastal shoreline }\end{array} & 46 \% & 53 \% \\\hline \text { Living farther inland } & 54 \% & 47 \% \\\hline\end{array}$$Graph the linear equations you wrote.
Use the table below, which gives the percents of people in the contiguous United States living within50 miles of a coastal shoreline and those living further inland.$$\begin{array}{|l|c|c|}\hline \text { Qummoninititicustics } & \text { 1940 } & \text { 1997 } \\\hline \begin{array}{l}\text { Living within 50 miles } \\\text { of a coastal shoreline }\end{array} & 46 \% & 53 \% \\\hline \text { Living farther inland } & 54 \% & 47 \% \\\hline\end{array}$$From your graphs, estimate when the percent of people living near the coast equaled the percent living inland.
You do 4 loads of laundry each week at a launderette where each load costs $\$ 1.25 .$ You could buy a washing machine that costs $\$ 400 .$ Washing 4 loads at home will cost about $\$ 1$ per week for electricity. How many loads of laundry must you do in order for the costs to be equal?
Which ordered pair is a solution of the linear system?$$\begin{aligned}&x+y=0.5\\&x+2 y=1\end{aligned}$$$\begin{array}{llll}\text { (A) }(0,-2) & \text { (B) }(-0.5,0) & \text { C } & (0,0.5)\end{array}$(D) $(0,-0.5)$
If $y-x=-3$ and $4 x+y=2,$ then $x=?$$$\begin{array}{lllll}\text { (A) } 1 & \text { (B) }-1 & \text { (C) } 5 & \text { (D) }-5\end{array}$$
You know how to solve the equation $\frac{1}{2} x+2=\frac{3}{2} x-12$ algebraically. This equation can also be solved graphically by solving the linear system. $$\begin{aligned}&y=\frac{1}{2} x+2\\&y=\frac{3}{2} x-12\end{aligned}$$a. Explain how the linear system is related to the original equation.b. Solve the system graphically.c. Check that the $x$ -coordinate from part (b) satisfies the original equation $\frac{1}{2} x+2=\frac{3}{2} x-12$ by substituting the $x$ -coordinate for $x$
Use the method shown in Exercise 43 to solve the equation graphically.$$\frac{4}{5} x=3 x-11$$
Use the method shown in Exercise 43 to solve the equation graphically.$$x+1=\frac{3}{2} x+2$$
Use the method shown in Exercise 43 to solve the equation graphically.$$1.2 x-2=3.4 x-13$$
Solve the equation.$$3 x+7=-2$$
Solve the equation.$$15-2 a=7$$
Solve the equation.$$21=7(w-2)$$
Solve the equation.$$2 y+5=3 y$$
Solve the equation.$$2(z-3)=12$$
Solve the equation.$$-(3-x)=-7$$
Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form.$$(3,0), m=-4$$
Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form.$$(-4,3), m=1$$
Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form.$$(1,-5), m=4$$
Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form.$$(-4,-1), m=-2$$
Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form.$$(2,3), m=2$$
Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form.$$(-1,5), m=\frac{2}{3}$$
The Verrazano-Narrows Bridge in New York City is the longest suspension bridge in North America, with a main span of 4260 feet. Write an inequality that describes the length in feet $x$ of every other suspension bridge in North America. Graph the inequality on a number line.