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Chapter 8: Problem 38

Solve each system by addition. Determine whether each system is independent, dependent, or inconsistent. $$\begin{aligned} &3 x-2 y=7\\\ &-3 x+y=5 \end{aligned}$$

Short Answer

Expert verified
x = -\frac{17}{3}, y = -12; independent

Step by step solution

01

- Align the Equations

Write the given system of equations in a clear format:\[\begin{aligned}3x - 2y &= 7 \-3x + y &= 5\end{aligned}\]
02

- Add the Equations

To eliminate one of the variables, add the two equations together:\[\begin{aligned}3x - 2y + (-3x + y) &= 7 + 5 \3x - 3x - 2y + y &= 12 \- y &= 12\end{aligned}\]
03

- Solve for y

Simplify the equation to solve for \(y\):\[-y = 12 \ y = -12\]
04

- Substitute y into One of the Original Equations

Choose the second equation to substitute \( y = -12 \):\[-3x + y = 5 \-3x - 12 = 5 \-3x = 17 \x = -\frac{17}{3}\]
05

- Verify the Solution

Substitute \( x = -\frac{17}{3} \) and \( y = -12 \) back into the original equations to verify:For the first equation:\[3\left(-\frac{17}{3}\right) - 2(-12) = 7 \-17 + 24 = 7 \7 = 7 \ \text{Correct}\]For the second equation:\[-3\left(-\frac{17}{3}\right) + (-12) = 5 \17 - 12 = 5 \5 = 5 \ \text{Correct}\]
06

- Classify the System

Since the solution \(x = -\frac{17}{3}\) and \( y = -12 \) satisfies both equations, the system is independent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Addition Method
The addition method, also known as the elimination method, is a powerful technique for solving systems of linear equations. It involves combining the equations in such a way that one of the variables is eliminated.

Here's a step-by-step breakdown:
  • Align the equations: Write both equations so that their corresponding terms line up vertically.
  • Add the equations: Combine like terms from both equations. This helps to eliminate one variable.
  • Solve for the remaining variable: Simplify the resulting equation to find the value of one variable.
  • Substitute back: Insert the found value into one of the original equations to solve for the other variable.
For example, in the given system:

\(\begin{aligned} 3x - 2y &= 7 \ -3x + y &= 5 \text{Adding these gives}\ -y &= 12 \ y &= -12 \ \text{We then solve for } x: \ \ -3x + (-12) = 5 \ x = - \frac {17}{3}\) This method is highly efficient and easy to learn with practice.
Dependent vs. Independent Systems
Understanding whether a system of equations is independent, dependent, or inconsistent is crucial in algebra.

  • Independent Systems: An independent system has exactly one unique solution. This means the lines representing the equations intersect at exactly one point. In our example, the system is independent because we found a unique solution: \(x = -\frac{17}{3}\) and \(y = -12\).
  • Dependent Systems: A dependent system has infinitely many solutions. This occurs when the equations represent the same line, meaning every point on the line is a solution. In such cases, after attempting to eliminate a variable, you'd end up with a true but redundant statement like \(0 = 0\).
  • Inconsistent Systems: An inconsistent system has no solution because the lines are parallel and never intersect. This will result in a false statement like \(0 = 5\) when attempting to solve the system.
Recognizing these types helps you understand the nature of the solutions and the relationships between the equations.
Verifying Solutions to Systems of Equations
Verifying solutions ensures the values found actually satisfy the original equations. Here鈥檚 how you can double-check your work:

  • Substitute the values: Take the solution pair \((x, y)\) and substitute the values back into both original equations.
  • Simplify both equations: Perform the indicated operations to see if both equations result in true statements.
For instance in our example: Substitute \(x = -\frac{17}{3}\) and \(y = -12\) into the original equations: \(\begin{aligned} 3\bigg(-\frac{17}{3}\bigg) - 2(-12) &= 7 \ -17 + 24 &= 7 \ 7 &= 7 --\Rightarrow \text{Correct!}\text{Similarly, for the second equation:}\-3\bigg(-\frac{17}{3}\bigg) + (-12) = 5 \ 17 - 12 = 5 \ 5 &= 5 \text{Right!}\) Both equations check out, confirming our solution is correct. Verifying solutions help eliminate errors and solidify understanding.

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Most popular questions from this chapter

Find a system of inequalities to describe the given region. Points that are closer to the \(x\) -axis than they are to the \(y\) -axis.

Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. Mr. Thomas and his three children paid a total of \(\$ 65.75\) for admission to Water World. Mr. and Mrs. Li and their six children paid a total of \(\$ 131.50 .\) What is the price of an adult's ticket and what is the price of a child's ticket?

Solve each system of inequalities. $$\begin{aligned}&x^{2}+y \leq 5\\\&y \geq x^{3}-x\\\&y \leq 4\end{aligned}$$

Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. Carmen made \(\$ 25,000\) profit on the sale of her condominium. She lent part of the profit to Jim's Orange Grove at \(10 \%\) interest and the remainder to Ricky's Used Cars at \(8 \%\) interest. If she received \(\$ 2200\) in interest after one year, then how much did she lend to each business?

Solve each system of inequalities. $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\\x+y \leq 4\end{array}$$
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