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Chapter 4: Problem 11

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

Short Answer

Expert verified
The solution set to the system of equations is {(x, y) | x = -2, y = -11/5}.

Step by step solution

01

Express one of the variables in terms of the other

Let's express 'x' in terms of 'y' using the second equation. That is, rearrange x - 2y = 4 to get x = 4 + 2y.
02

Substitute x into the other equation

Substitute x from step 1 into the first equation, and solve for y. We get -3(4+2y) + y = -1. This simplifies to -12 - 6y + y = -1, which further simplifies to -5y - 12 = -1. Adding 12 to both sides gives -5y = 11. Dividing by -5 results with y = -11/5.
03

Substitute y into x to get its value

Substitute y = -11/5 back into the equation for x in step 1 to get x = 4 + 2(-11/5). Simplifies to x = 4 - 22/5 = -2.
04

Express the solution set in set notation

The solution set in set notation will be given as {(x, y) | x = -2, y = -11/5}. This is an ordered pair representing the solution to the system of equations.

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Most popular questions from this chapter

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

Involve dual investments. Your grandmother needs your help. She has 50,000 dollar to invest. Part of this money is to be invested in noninsured bonds paying \(15 \%\) annual interest. The rest of this money is to be invested in a government-insured certificate of deposit paying \(7 \%\) annual interest. She told you that she requires a total of 6000 dollar per year in extra income from these investments. How much money should be placed in each investment?

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

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

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. $$\left\\{\begin{array}{l} 4(3 x-y)=0 \\ 3(x+3)=10 y \end{array}\right.$$
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