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Chapter 4: Problem 48

Solve by substitution. $$ \begin{aligned} &w=\frac{3}{8} v+5 \\ &-3 v+8 w=41 \end{aligned} $$

Short Answer

Expert verified
There is no solution.

Step by step solution

01

- Solve one equation for one variable

Given the equation: \[ w=\frac{3}{8}v+5 \], solve for \( w \).
02

- Substitute into the second equation

Substitute \( w = \frac{3}{8}v + 5 \) into the second equation: \(-3v + 8w = 41\). This gives us: \(-3v + 8\left(\frac{3}{8}v + 5\right) = 41\).
03

- Simplify the equation

Simplify the equation: \[ -3v + 8\left(\frac{3}{8}v + 5\right) = 41 \]\[ -3v + 3v + 40 = 41 \]\[ 40 = 41 \] This shows a contradiction, indicating that there is no solution that satisfies both equations simultaneously.

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Key Concepts

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

substitution method
The substitution method is a technique for solving systems of equations. It involves solving one of the equations for one variable and then substituting that solution into the other equation. Let's dive into its application to our exercise step-by-step.
First, we solve the equation: \( w=\frac{3}{8}v+5 \) for the variable [w].
Next, we substitute this value of \( w \) into the second equation, \( -3v + 8w = 41 \). This means we replace every instance of \( w \) in the equation with \( \frac{3}{8}v + 5 \).
So, \[ -3v + 8\bigg(\frac{3}{8}v + 5\bigg) = 41 \]
When we distribute and simplify, it gives us a clearer picture of the relationship between [v] and [w].
consistent and inconsistent systems
Consistent systems have at least one solution, while inconsistent systems have no solution. After substituting, we simplify the equation: \[ -3v + 3v + 40 = 41 \]
All terms involving \( v \) cancel out, leaving: \[ 40 = 41 \]
This is a contradiction, which shows no values for \[ v \] and \[ w \] can satisfy both equations simultaneously.
Hence, the system is inconsistent as no solution exists.
elementary algebra problems
In elementary algebra, solving systems of equations is a basic but essential skill. These systems can either be linear or nonlinear. This exercise involves two linear equations.
The process usually involves:
  • Identifying variables in each equation.
  • Isolating one variable to simplify substitution.
  • Substituting and simplifying as shown.

Understanding the nature (consistent or inconsistent) of the systems helps in recognizing if there's a solution or not.
This approach not only solves the problem but also fosters a deeper understanding of algebraic principles.

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Most popular questions from this chapter

A total of $$\$ 16,000$$ is invested in two mutual funds for 1 year. The return on Mutual Fund \(\mathrm{A}\) is \(2 \%\) per year, the return on Mutual Fund B is \(3 \%\) per year, and the total return is \(\$ 441.50\). Find the amount invested in Mutual Fund A and the amount invested in Mutual Fund B.

A campsite is \(6 \mathrm{mi}\) from a trailhead. One group of hikers begins at the trailhead, hikes on the trail to this campsite, and spends the night. At 8 a.m. the next morning, they start hiking on the trail again at a speed of 3 mi per hour. At 10 a.m. the same day, another group of hikers begins the trail at the trailhead, hiking at a speed of 5 mi per hour. Find the time and distance hiked by the second group when they catch up with the first group. (Assume that the groups do not take breaks or stop.)

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