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Chapter 3: Problem 31

If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this. Solve using the elimination method. $$ \begin{aligned} &9 x+3 y=-3\\\ &2 x-3 y=-8 \end{aligned} $$

Short Answer

Expert verified
The system of equations has a unique solution: \( x = -1 \) and \( y = 2 \).

Step by step solution

01

- Add the Two Equations

First, we add the two given equations to eliminate the variable y. The given equations are: \ \( 9x + 3y = -3 \) and \( 2x - 3y = -8 \). By adding them together, we get: \ \[ (9x + 3y) + (2x - 3y) = -3 + (-8) \] \ The terms with y cancel out, so: \ \[ 11x = -11 \]
02

- Solve for x

Next, solve for x by dividing both sides of the equation by 11: \ \[ x = -1 \]
03

- Substitute x Back into One of the Original Equations

Substitute \( x = -1 \) back into either of the original equations to solve for y. Using the first equation: \ \[ 9(-1) + 3y = -3 \] \ \[ -9 + 3y = -3 \] \ \[ 3y = 6 \] \ \[ y = 2 \]
04

- Write the Solution

The solution to the system of equations is \( x = -1 \) and \( y = 2 \). Since we found a single unique solution, the system has exactly one solution.

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Key Concepts

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

Elimination Method
The elimination method is a widely-used technique for solving systems of linear equations. The goal is to eliminate one variable by adding or subtracting equations, making it easier to solve for the remaining variables.

Here's how you can use the elimination method effectively:
  • Make sure the coefficients of one of the variables are opposites. If they are not, you may need to multiply one or both equations by a constant to make this happen.
  • Add or subtract the equations to eliminate one of the variables.
  • Solve for the remaining variable.
  • Substitute the solved variable back into one of the original equations to find the value of the other variable.
For example, with equations: \( 9x + 3y = -3 \) and \( 2x - 3y = -8 \), adding them cancels out y, making it possible to solve for x.
Unique Solution
A system of linear equations can have either a unique solution, no solution, or infinitely many solutions. A unique solution means the system intersects at exactly one point.

In our problem, after using the elimination method, we found: \[ 11x = -11 \] Solving for x, we get: \[ x = -1 \]
Substituting \( x = -1 \) back into the first equation:
\[ 9(-1) + 3y = -3 \]
we find y: \[ -9 + 3y = -3 \] \[ 3y = 6 \] \[ y = 2 \]
Thus, the unique solution is \( x = -1 \) and \( y = 2 \). The system has exactly one unique solution.
Set-Builder Notation
Set-builder notation is a mathematical notation used to describe a set by specifying the properties that its members must satisfy. It鈥檚 very useful for defining solution sets of equations, especially when there are infinitely many solutions.
For example, if a system of linear equations has infinitely many solutions, we might write the solution set using set-builder notation:
\[ \begin{align*} S = \{ (x, y) \mid 9x + 3y = -3 \} \end{align*} \] This notation tells us that any \( x \) and \( y \) values which satisfy the equation \( 9x + 3y = -3 \) belong to the set S.
It's a compact and precise way to represent all possible solutions.
Substitution
The substitution method is another popular technique for solving systems of equations. Here, one equation is solved for one variable in terms of the other variables, and this expression is substituted into the other equation.

To use substitution:
  • Solve one of the equations for one variable in terms of the other.
  • Substitute this expression into the other equation.
  • Solve for the remaining variable.
  • Substitute back to find the other variable.
For instance, if we solve \( 9x + 3y = -3 \) for \( y \): \[ 3y = -3 - 9x \] \[ y = -1 - 3x \]
Then substitute \( y \) into the second equation:
\[ 2x - 3(-1 - 3x) = -8 \]
Solving this will yield the values of \( x \) and then substituting that value to find \( y \).

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Most popular questions from this chapter

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Determine whether \((2,-1,-2)\) is a solution of the system $$ \begin{aligned} x+y-2 z &=5 \\ 2 x-y-z &=7 \\ -x-2 y-3 z &=6 \end{aligned} $$
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