Chapter 4

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}y=\frac{1}{2} x-3 \\ y=\frac{1}{2} x-5\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}y \geq 2 x+2 \\\y<2 x-3 \\\x \geq 2\end{array}\right.$$

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} \frac{x}{3}+y=3 \\ \frac{x}{2}-\frac{y}{4}=1 \end{array}\right.$$

Explain how to solve a system of equations using the substitution method. Use \(y=3-3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}y=\frac{3}{4} x-2 \\ y=\frac{3}{4} x+1\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}y \geq-3 x+2 \\\y<-3 x \\\x \geq 1\end{array}\right.$$

Exercises \(17-20\) involve using systems of linear equations to compare costs of telephone plans and plans at a discount warehouse. Describe another situation that involves choosing between two options that can be modeled and solved with a linear system.

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 3 x-2 y &=8 \\ x &=-2 y \end{aligned}\right.$$

When using the substitution method, how can you tell if a system of linear equations has no solution?

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}y=-\frac{1}{2} x+4 \\ 3 x-y=-4\end{array}\right.$$

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 2 x-y &=10 \\ y &=3 x \end{aligned}\right.$$

When using the substitution method, how can you tell if = system of linear equations has a infinitely many solutions?

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}y=-\frac{1}{4} x+3 \\ 4 x-y=-3\end{array}\right.$$

Must the concentration of a mixture always be greater than the concentration of an ingredient in one of the solutions and less than the concentration of the ingredient in the other solution being mixed? Explain your answer.

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} 3 x+2 y=-3 \\ 2 x-5 y=17 \end{array}\right.$$

The law of supply and demand states that, in a free markes economy, a commodity tends to be sold at its equilibriunprice. At this price, the amount that the seller will supply is the same amount that the consumer will buy. Explair how systems of equations can be used to determine the equilibrium price.

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}3 x-y=6 \\ x=\frac{y}{3}+2\end{array}\right.$$

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} 2 x-7 y=17 \\ 4 x-5 y=25 \end{array}\right.$$

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 3 x-2 y &=6 \\ y &=3 \end{aligned}\right.$$

When using substitution to solve $$\left\\{\begin{array}{rr}5 x-4 y= & 9 \\\x-2 y= & -3\end{array}\right.$$ I find it easiest to solve for \(x\) in the first equation.

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}3 x+y=0 \\ y=-3 x+1\end{array}\right.$$

Suppose a patient is not allowed to have more than 330 milligrams of cholesterol per day from a diet of eggs and meat. Each egg provides 165 milligrams of cholesterol and each ounce of meat provides 110 milligrams of cholesterol. Thus, \(165 x+110 y \leq 330,\) where \(x\) is the number of eggs and \(y\) the number of ounces of meat. Furthermore, the patient must have at least 165 milligrams of cholesterol from the diet. Graph the system of inequalities in the first quadrant. Give the coordinates of any two points in the solution set. Describe what each set of coordinates means in terms of the variables in the problem.

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 2 x+3 y &=7 \\ x &=2 \end{aligned}\right.$$

Find the slope and the \(y\) -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\\{\begin{array}{l}2 x+y=0 \\ y=-2 x+1\end{array}\right.$$

What does the graph of a system of linear inequalities represent?

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} y=2 x+1 \\ y=2 x-3 \end{array}\right.$$

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} y=2 x+4 \\ y=2 x-1 \end{array}\right.$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Solving an inconsistent system by substitution results in a true statement.

A band plans to record a demo. Studio A rents for \(\$ 100\) plus \(\$ 50\) per hour. Studio \(B\) rents for \(\$ 50\) plus \(\$ 75\) per hour. The total cost, \(y,\) in dollars, of renting the studios for \(x\) hours can be modeled by the linear system $$\left\\{\begin{array}{l}y=50 x+100 \\ y=75 x+50\end{array}\right.$$ a. Use graphing to solve the system. Extend the \(x\) -axis from 0 to 4 and let each tick mark represent 1 unit (one hour in a recording studio). Extend the \(y\) -axis from 0 to 400 and let each tick mark represent 100 units (a rental cost of \(\$ 100\) ). b. Interpret the coordinates of the solution in practical terms.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The reason that systems of linear inequalities are appropriate for modeling healthy weight is because guidelines give healthy weight ranges, rather than specific weights, for various heights.

Tourist: "How many birds and lions do you have in your zoo?" Zookeeper: "There are 30 heads and 100 feet." Tourist: "I can't tell from that." Zookeeper: "Oh, yes, you can!" Can you? Find the number of each.

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 2(x+2 y) &=6 \\ 3(x+2 y-3) &=0 \end{aligned}\right.$$

You plan to start taking an aerobics class. Nonmembers pay \(\$ 4\) per class. Members pay a \(\$ 10\) monthly fee plus an additional \(\$ 2\) per class. The monthly cost, \(y,\) of taking \(x\) classes can be modeled by the linear system $$\left\\{\begin{array}{l}y=4 x \\ y=2 x+10\end{array}\right.$$ a. Use graphing to solve the system. b. Interpret the coordinates of the solution in practical terms.

The reason that systems of linear inequalities are appropriate for modeling healthy weight is because guidelines give healthy weight ranges, rather than specific weights, for various heights. I graphed the solution set of \(y \geq x+2\) and \(x \geq 1\) without using test points.

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} 2(x+y)=4 x+1 \\ 3(x-y)=x+y-3 \end{array}\right.$$

What is a system of linear equations? Provide an example with your description.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I graphed the solution set of \(2 x-y<4\) and \(x+y>-1\) without using test points.

One apartment is directly above a second apartment. The resident living downstairs calls his neighbor living above him and states, "If one of you is willing to come downstairs, we'll have the same number of people in both apartments." The upstairs resident responds, "We're all too tired to move. Why don't one of you come up here? Then we'll have twice as many people up here as you've got down there." How many people are in each apartment?

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 3 y &=2 x \\ 2 x+9 y &=24 \end{aligned}\right.$$

Determine whether each statement is true or false. If the statement is false, make the necessary To solve the system $$\left\\{\begin{array}{l}2 x-y=5 \\\3 x+4 y=7\end{array}\right.$$ Eby substitution, replace \(y\) in the second equation with \(5-2 x\)

What is a solution of a system of linear equations?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I use two different colors to graph solution sets of systems of inequalities, selecting the region where the colors overlap.

In Lewis Carroll's Through the Looking Glass, the Following dialogue takes place: Tweedledum (to Tweedledee): The sum of your weight and twice mine is 361 pounds. Tweedledee (to Tweedledum): Contrawise, the sum of your weight and twice mine is 362 pounds. Find the weight of each of the two characters. (IMAGE CAN NOT COPY)

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 4 y &=-5 x \\ 5 x+8 y &=20 \end{aligned}\right.$$

If \(x=3-y-z, 2 x+y-z=-6,\) and $$3 x-y+z=11,\( find the values for \)x, y,\( and \)z$$

Explain how to determine if an ordered pair is a solution of a system of linear equations.

You have 70,000 dollar to invest. Part of the money is to be placed in a certificate of deposit paying \(8 \%\) per year. The rest is to be placed in corporate bonds paying \(12 \%\) per year. If you wish to obtain an overall return of \(9 \%\) per year, how much should you place in each investment?

In Exercises \(57-60\), write a system of equations modeling the given conditions. Then solve the system by the addition method and find the two numbers. Five times a first number increased by a second number is \(14 .\) The difference between four times the first number and the second number is \(4 .\) Find the numbers.

Find the value of \(m\) that makes $$\left\\{\begin{array}{l}y=m x+3 \\\5 x-2 y=7\end{array}\right.$$ an inconsistent system.

Explain how to solve a system of linear equations by graphing.