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Chapter 3: Problem 25

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. $$ \begin{aligned} &2 x-3=y\\\ &y-2 x=1 \end{aligned} $$

Short Answer

Expert verified
The system has no solution.

Step by step solution

01

Solve one equation for one variable

Choose one of the equations and solve it for one of the variables. Looking at the first equation, solve for y: \[ 2x - 3 = y \] This can be rewritten as: \[ y = 2x - 3 \]
02

Substitute the expression into the other equation

Now, substitute the expression for y from the first equation into the second equation: \[ y - 2x = 1 \] Substituting \( y = 2x - 3 \), we get: \[ (2x - 3) - 2x = 1 \]
03

Simplify and solve for x

Simplify the equation: \[ 2x - 3 - 2x = 1 \] Which simplifies to: \[ -3 = 1 \] This is a contradiction because -3 does not equal 1.
04

Determine the type of solution

Since the simplified equation resulted in a contradiction, the system has no solution. The two lines are parallel and do not intersect at any point.

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Key Concepts

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

The Substitution Method
The substitution method is a way to solve systems of linear equations by substituting one equation into the other. This is a step-by-step process that simplifies the system to find the values of the variables.

Here鈥檚 how it works:
  • Solve one of the equations for one of the variables.
  • Substitute this expression into the other equation.
  • Simplify and solve for the remaining variable.
  • Finally, substitute back to find the value of the first variable.

In the given example, we first solved for y in the first equation:
\[ y = 2x - 3 \]
Then, we substituted this expression into the second equation. Upon simplifying, we encountered a contradiction: \[ -3 = 1 \]
This tells us the system has no solution.
Inconsistent System
An inconsistent system of linear equations is one that has no solutions. This occurs when the lines represented by the equations do not intersect. The result is a contradiction, indicating there are no points in common.

In our example, after substituting and simplifying, we found \[ -3 = 1 \], which is clearly false. This contradiction shows that our system is inconsistent.
  • When simplifying leads to a false statement, the system is inconsistent.
  • An inconsistent system means the lines never meet, hence, no solutions.

Understanding how to identify inconsistent systems helps in determining the solution or lack thereof for a set of linear equations.
Parallel Lines
When discussing linear equations, it's essential to understand the concept of parallel lines. Lines are parallel if they never intersect, which means they have the same slope but different y-intercepts.

In our example:
  • The equations are \[ y = 2x - 3 \] and \[ y = 2x + 1 \].
  • Both lines have a slope of 2 but different y-intercepts ( -3 and 1 ).
  • This confirms the lines are parallel and do not intersect.

Thus, the parallel nature of these lines signifies that our system is inconsistent. Recognizing parallel lines in systems of linear equations can quickly tell us if a solution doesn't exist.

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Most popular questions from this chapter

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The ACT is a standardized test for students entering college. Each of the four scores that a student receives has a benchmark value. Students scoring at or above the benchmarks are considered ready to succeed in college. The benchmark for the science test is 6 points higher than the benchmark for the English test. The sum of the reading and mathematics benchmarks is 1 point more than the sum of the English and science benchmarks. The sum of the English, mathematics, and science benchmarks is 1 point more than three times the reading benchmark. The sum of all four benchmarks is \(85 .\) Find all four benchmarks.

In each case, three solutions of an equation in \(x, y,\) and \(z\) are given. Find the equation. $$ \begin{aligned} &z=b-m x-n y\\\ &(1,1,2),(3,2,-6), \text { and }\left(\frac{3}{2}, 1,1\right) \end{aligned} $$
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