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Chapter 3: Problem 33

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. $$ \begin{aligned} 5 x+3 y &=19 \\ x-6 y &=11 \end{aligned} $$

Short Answer

Expert verified
The solution set is \( \left( \frac{49}{11}, -\frac{12}{11} \right) \).

Step by step solution

01

Multiply to Eliminate One Variable

To eliminate one variable, multiply the two equations by appropriate values so that the coefficients of one of the variables become equal. Multiply the second equation by 5 to align the coefficient of x with the first equation:\[5(x - 6y) = 5(11)\]This simplifies to:\[5x - 30y = 55\]
02

Set Up Eliminated Equations

Write the modified equations together:\[5x + 3y = 19 \5x - 30y = 55\]
03

Subtract the Equations

Subtract the second equation from the first to eliminate \(5x\):\[(5x + 3y) - (5x - 30y) = 19 - 55\]Simplifies to:\[33y = -36\]
04

Solve for y

Solve for \(y\) in the simplified equation:\[33y = -36\]\[y = \frac{-36}{33}\]\[y = -\frac{12}{11}\]
05

Substitute Back to Find x

Substitute \(y = -\frac{12}{11}\) back into one of the original equations to find \(x\). Using the second equation:\[x - 6\left(-\frac{12}{11}\right) = 11\]\[x + \frac{72}{11} = 11\]\[x = 11 - \frac{72}{11}\]\[x = \frac{121}{11} - \frac{72}{11}\]\[x = \frac{49}{11}\]
06

Write the Solution

The pair \(\left( \frac{49}{11}, -\frac{12}{11} \right)\) is the unique solution.

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Key Concepts

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

system of equations
A system of equations is a set of two or more equations that have common variables. The main goal when solving these systems is to find the value(s) of the variable(s) that satisfy all equations simultaneously. There are several methods to solve systems of equations, including substitution, graphing, and elimination. In the elimination method, we manipulate the equations to cancel out one of the variables, making it easier to solve for the remaining variable.
infinite solutions
Sometimes, when solving a system of equations, you might find that there are infinite solutions. This occurs when the equations represent the same line, meaning they overlap completely on a graph. In such cases, every point on the line is a solution. When using the elimination method, you might end up with a tautology like \(0 = 0\), indicating infinite solutions. For example, if after elimination, the simplified equations are identical, it shows that any \(x, y\) pair on the line is a solution.
set-builder notation
If a system has infinite solutions, we use set-builder notation to express the solution set. Set-builder notation is a way of describing a set by stating a property that its members must satisfy. For instance, if the system \(3x + 2y = 6\) and \(6x + 4y = 12\) lead to infinite solutions, we can represent the solution set as \{(x, y) | 3x + 2y = 6\}. This notation indicates that any pair (x, y) that satisfies the given equation is part of the solution set.
no solution
A system of equations might also have no solution. This happens when the equations represent parallel lines that never intersect. When using the elimination method, if you end up with a contradiction, such as \(0 = 5\), it indicates that no solutions exist. This means there are no pairs \(x, y\) that will satisfy both equations simultaneously. In such cases, we simply state that the system has no solution, reflecting that the lines do not cross at any point.

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