Chapter 4

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. Three times a first number increased by twice a second number is \(11 .\) The difference between the first number and twice the second number is 9. Find the numbers.

Graph: \(4 x+6 y=12 .\)

What is an inconsistent system? What happens if you attempt to solve such a system by graphing?

Write a system of inequalities that has no solutions.

$$\text { Solve: } 2(x+3)=24-2(x+4)$$

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. If four times a first number is decreased by three times a second number, the result is \(0 .\) The sum of the numbers is \(-7 .\) Find the numbers.

Solve: 4(x+1)=25+3(x-3)

Explain how a linear system can have infinitely many solutions.

Write a system of inequalities that describes all the points in quadrant III of a rectangular coordinate system.

Simplify: \(5+6(x+1)\)

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. If three times a first number is decreased by six times a second number, the result is \(15 .\) The sum of the numbers is \(2 .\) Find the numbers.

List all the integers in this set: $$\left\\{-73,-\frac{2}{3}, 0, \frac{3}{1}, \frac{3}{2}, \frac{\pi}{1}\right\\}$$

What are dependent equations? Provide an example with your description.

A person plans to invest no more than \(\$ 15,000,\) placing the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least \(\$ 2000\) is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Write and graph a system of inequalities that describes all possibilities for placing the money in the high- and low- risk investments.

Write the slope-intercept form of the equation of the Tine passing through \((-5,6)\) and \((3,-10)\)

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{3 x}{5}+\frac{4 y}{5}=1 \\ \frac{x}{4}-\frac{3 y}{8}=-1 \end{array}\right.$$

Will help you prepare for the material covered in the next section. Use both equations in the system $$\left\\{\begin{array}{l}3 x+2 y=48 \\\9 x-8 y=-24\end{array}\right.$$ to find \(x\) for \(y=12 .\) What do you observe?

The following system models the winning times, \(y,\) in seconds, in the Olympic 500 -meter speed skating event \(x\) years after 1970: $$\left\\{\begin{array}{l}y=-0.19 x+43.7 \\ y=-0.16 x+39.9\end{array}\right.$$ Use the slope of each model to explain why the system has a solution. What does this solution represent?

Promoters of a rock concert must sell at least \(25,000\) tickets priced at \(\$ 35\) and \(\$ 50\) per ticket. Furthermore, the promoters must take in at least \(\$ 1,025,000\) in ticket sales. Write and graph a system of inequalities that describes all possibilities for selling the \(\$ 35\) tickets and the \(\$ 50\) tickets.

Graph the given inequality in a rectangular coordinate system. $$2 x-y<4$$

Solve $$-14 y=-168$$

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{3}-\frac{y}{2}=\frac{2}{3} \\ \frac{2 x}{3}+y=\frac{4}{3} \end{array}\right.$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. Each equation in a system of linear equations has infinite many ordered-pair solutions.

Find the slope of the line containing the points \((-6,1)\) and \((2,-1) .\) (Section 3.3, Example 1)

Graph the given inequality in a rectangular coordinate system. $$y \geq x+1$$

Multiply both sides of \(x-5 y=3\) by \(-4\) and simplify.

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} 5(x+1)=7(y+1)-7 \\ 6(x+1)+5=5(y+1) \end{array}\right.$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. Every linear system has infinitely many ordered-pair solutions.

Add: \(\frac{1}{5}+\left(-\frac{3}{4}\right) .\) (Section 1.5, Example 4)

Graph the given inequality in a rectangular coordinate system. $$x \geq 2$$

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} 6 x=5(x+y+3)-x \\ 3(x-y)+4 y=5(y+1) \end{array}\right.$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. In dependent systems, the two equations represent the same line.

Solve: \(7 x=10+6(11-2 x)\) (Section \(2.3,\) Example 3 )

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} 0.4 x+y=2.2 \\ 0.5 x-1.2 y=0.3 \end{array}\right.$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. When I use graphing to solve an inconsistent system, the lines should look parallel, and I can always use slope to confirm that they really are.

Will help you prepare for the material covered in the first section of the next chapter. $$\text { Add: } 5 x^{3}+12 x^{3}$$

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{aligned} 1.25 x-1.5 y &=2 \\ 3.5 x-1.75 y &=10.5 \end{aligned}\right.$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a linear system has graphs with equal slopes, the system must be inconsistent.

Will help you prepare for the material covered in the first section of the next chapter. $$\text { Add: }-8 x^{2}+6 x^{2}$$

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{2}=\frac{y+8}{3} \\ \frac{x+2}{2}=\frac{y+11}{3} \end{array}\right.$$

Will help you prepare for the material covered in the first section of the next chapter. $$\text { Subtract: }-9 y^{4}-\left(-2 y^{4}\right)$$

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{2}=\frac{y+8}{4} \\ \frac{x+3}{2}=\frac{y+5}{4} \end{array}\right.$$

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Write a system of equations with one solution, a system of equations with no solution, and a system of equations with infinitely many solutions. Explain how you were able to think of these systems.

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

Verify your solutions to any five exercises from Exercises 11 through 36 by using a graphing utility to graph the two equations in the system in the same viewing rectangle. After entering the two equations, one as \(y_{1}\) and the other as \(y_{2},\) and graphing them, use the \([\text { INTERSECTION }]\) feature to find the system's solution. (It may first be necessary to solve the equations for \(y\) before entering them.)

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

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}y=2 x+2 \\ y=-2 x+6\end{array}\right.$$

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}y=-x+5 \\ y=x-7\end{array}\right.$$

The formula \(3239 x+96 y=134,014\) models the number of daily evening newspapers, \(y, x\) years after \(1980 .\) The formula \(-665 x+36 y=13,800\) models the number of daily morning newspapers, \(y, x\) years after \(1980 .\) What is the most efficient method for solving this system? Explain why. What does the solution mean in terms of the variables in the formulas? (It is not necessary to actually solve the system.)