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Chapter 4: Problem 9

(a) solve by substitution. (b) if there is one solution, check. $$ \begin{aligned} &y=4 x-1 \\ &y=4 x+3 \end{aligned} $$

Short Answer

Expert verified
There is no solution.

Step by step solution

01

- Equate the equations

Given the equations are: 1) \( y = 4x - 1 \)2) \( y = 4x + 3 \)Since both expressions equal \( y \), set the right sides equal to each other:en\( 4x - 1 = 4x + 3 \).
02

- Isolate variable

Subtract \( 4x \) from both sides to isolate the constant terms:en\( -1 = 3 \).
03

- Understand the result

The result \( -1 = 3 \) is a contradiction. This indicates that there is no solution where both equations are true simultaneously.

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Key Concepts

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

solving equations
Solving equations is a foundational skill in algebra. It involves finding the value of unknown variables that make the equation true. In simple terms, if you have an equation like \( y = 4x - 1 \), you want to find the value of \( x \) that satisfies the equation.
One of the most effective methods to solve equations is by substitution. This means you replace one variable with an expression containing the other variable.
For example, if you have two equations: \( y = 4x - 1 \) and \( y = 4x + 3 \), you substitute one equation into the other. Here, since both equal \( y \), you set the expressions equal to each other: \( 4x - 1 = 4x + 3 \).
system of equations
A system of equations involves two or more equations with the same set of unknowns. Often, you need to find the values of these unknowns that satisfy all the equations simultaneously.
There are several methods to solve systems of equations, including graphing, substitution, and elimination.
In our provided example, we use substitution because it's straightforward when an equation is already solved for one variable.
By substituting the expressions for \( y \) from both equations \( y = 4x - 1 \) and \( y = 4x + 3 \), we find a contradiction. This indicates there is no common solution.
no solution
When solving systems of equations, encountering a contradiction such as \( -1 = 3 \) is a sign that there is no solution.
This happens when the equations represent parallel lines that never intersect. Each line does not share any common point.
In algebraic terms, if after substitution or elimination, the variables cancel out leaving a false statement, it means the system has no solution. Identifying no solutions is crucial because it tells you that it's impossible to find values for the variables that satisfy all equations in the system.

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Most popular questions from this chapter

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For exercises \(85-88\), the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Write a system of equations that represents the relationship of the variables. Mixture A is 6\% juice. Mixture B is \(15 \%\) juice. Mixture B contains \(50 \%\) of the RDA of vitamin C. Find the amount of each mixture needed to make 80 gal of a new drink that is \(11 \%\) juice. Round to the nearest whole number. $$ \begin{aligned} &x=\text { amount of Mixture A } \\ &y=\text { amount of Mixture B } \\ &\text { Incorrect Answer: } x+y=80 \text { gal } \\ &\qquad 0.06 x+0.50 y=(0.11)(80 \mathrm{gal}) \end{aligned} $$

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