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Chapter 3: Problem 15

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 substitution method. $$ \begin{aligned} &3 s-4 t=14\\\ &5 s+t=8 \end{aligned} $$

Short Answer

Expert verified
The solution is \( s = 2 \) and \( t = -2 \).

Step by step solution

01

- Solve one equation for one variable

Choose the second equation, which is easier to solve for one of the variables. Solve for t: \[ 5s + t = 8 \] Subtract \(5s\) from both sides: \[ t = 8 - 5s \]
02

- Substitute into the other equation

Substitute \(t = 8 - 5s\) into the first equation: \[ 3s - 4(8 - 5s) = 14 \]
03

- Simplify the equation

Distribute \(-4\): \[ 3s - 32 + 20s = 14 \]Combine like terms: \[ 23s - 32 = 14 \]
04

- Solve for s

Add 32 to both sides: \[ 23s = 46 \]Divide by 23: \[ s = 2 \]
05

- Solve for t using the value of s

Use the equation \( t = 8 - 5s \) and substitute \( s = 2 \): \[ t = 8 - 5(2) = 8 - 10 = -2 \]
06

- State the solution

The solution to the system of equations is \( s = 2 \) and \( t = -2 \). There is a single unique solution.

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Key Concepts

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

Solving Systems of Equations
Solving systems of equations involves finding the values of variables that satisfy multiple equations simultaneously.
In this exercise, we have two equations involving variables s and t:
  • 3s - 4t = 14
  • 5s + t = 8
When we solve these equations, we need to determine the values of s and t that make both equations true at the same time.
The substitution method is a common technique used to solve systems of equations. We solve one of the equations for one of the variables and then substitute that expression into the other equation.
In this case, we solve the second equation for t, and then substitute that expression for t into the first equation.
Set-Builder Notation
Set-builder notation is a mathematical way of describing a set by stating the properties that its members must satisfy.
It is especially useful for describing solution sets that have infinite solutions.
If a system of equations has an infinite number of solutions, we can describe the solution set using set-builder notation.
For example, if s and t had infinite solutions, we could use set-builder notation like this: \[ \{ (s,t) | t = 8 - 5s \}\]
However, in our case, the system has a unique solution, so set-builder notation is not necessary.
We just write the single solution as \[ (s, t) = (2, -2)\].
Unique Solutions
When solving systems of equations, there are three possible outcomes: a unique solution, no solution, or infinitely many solutions.
  • A unique solution occurs when the system of equations intersects at a single point.
  • No solution happens when the equations represent parallel lines that never intersect.
  • Infinite solutions occur when the equations represent the same line, so every point on the line is a solution.

In our given system of equations, we found a unique solution of s = 2 and t = -2.
This means the two lines intersect at one specific point.The process of substitution leads us to this single intersection point, ensuring both equations are satisfied.

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Most popular questions from this chapter

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The tens digit of a two-digit positive integer is 2 more than three times the units digit. If the digits are interchanged, the new number is 13 less than half the given number. Find the given integer. (Hint. Let \(x=\) the tens-place digit and \(y=\) the units-place digit; then \(10 x+y\) is the number.

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