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Chapter 4: Problem 20

(a) solve by elimination. (b) if there is one solution, check. $$ \begin{aligned} &x+y=6 \\ &x+y=1 \end{aligned} $$

Short Answer

Expert verified
There is no solution.

Step by step solution

01

Write the system of equations

The given system of equations is: \(x + y = 6\) \(x + y = 1\)
02

Subtract the second equation from the first

Subtract the second equation from the first to eliminate one of the variables:\((x + y) - (x + y) = 6 - 1\) This simplifies to: \(0 = 5\)
03

Analyze the result

The result \(0 = 5\) is a contradiction, which means there is no solution to the system of equations.
04

Conclude there is no solution

Since the result is a contradiction, we conclude that there is no solution to the system using elimination.

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Key Concepts

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

elimination method
The elimination method is a technique used to solve systems of linear equations. The main goal is to eliminate one of the variables by adding or subtracting the equations.

To start, align the given equations so that corresponding variables and constants match in columns. Here is how it works:
  • Given the system: \(x + y = 6\) and \(x + y = 1\), these equations are already aligned.
  • Next step is to add or subtract the two equations to eliminate one of the variables.
  • In this case, we subtract the second equation from the first: \((x + y) - (x + y)\), which simplifies to \(0\).

    • We also subtract the constants on the right side of the equations: \(6 - 1 = 5\).

    Finally, you end up with \(0 = 5\). This shows that a contradiction has been reached.
contradiction in equations
A contradiction in equations means we have arrived at an impossible statement. This typically arises in a system of linear equations when two lines are parallel and never intersect.

In our example, the contradiction was \(0 = 5\). Here's why this happens:
  • The given equations \(x + y = 6\) and \(x + y = 1\) suggest that adding the same two variables would give two different sums, which is not possible.
  • This means the lines represented by these equations have the same slope but different y-intercepts.
Therefore, there is no point (x, y) that satisfies both equations simultaneously.
no solution systems
When solving systems of linear equations, it is possible to find that no solutions exist. No solution systems occur when the lines represented by the equations are parallel and distinct.

We identify no solution systems by:
  • Finding contradictory statements like \(0 = 5\) when using elimination or substitution methods.
  • Recognizing that the equations have the same slope but different y-intercepts.
  • Understanding that parallel lines never intersect, hence no single point can satisfy both equations.

It鈥檚 essential to recognize the signs of no solution to avoid wasting time attempting to find a non-existent solution. In our example, the system \(x + y = 6\) and \(x + y = 1\) clearly has no solution.

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Most popular questions from this chapter

A quilter has 108 yd of fabric. She wants to donate five times as much fabric to the Quilts for Children project as she keeps. Find the number of yards that she should donate. Find the number of yards that she should keep.

The perimeter of a rectangle is 46 in. The length plus three times the width is \(37 \mathrm{in}\). Find the length and width.

The cost of regular gasoline is $$\$ 3.85$$ per gallon and the cost of premium gasoline is $$\$ 4.05$$ per gallon. If a total of \(1200 \mathrm{gal}\) of gasoline is sold for \(\$ 4675\), found the amount of regular gasoline and the amount of premium gasoline sold.

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. Each pound of Cereal A contains \(0.5 \mathrm{oz}\) of almonds and \(2 \mathrm{oz}\) of dried cranberries. Each pound of Cereal B contains 2 oz of almonds and 1 oz of dried cranberries. Find the amount of each cereal needed to make 400 lb of a new cereal that contains \(0.75 \mathrm{oz}\) of almonds per pound of cereal. (1 \(\mathrm{lb}=16 \mathrm{oz}\).) $$ \begin{aligned} &x=\text { amount of Cereal A } \\ &y=\text { amount of Cereal B } \end{aligned} $$ Incorrect Answer: \(x+y=400 \mathrm{lb}\) $$ \left(\frac{0.5 \mathrm{oz}}{1 \mathrm{lb}}\right) x+\left(\frac{2 \mathrm{oz}}{1 \mathrm{lb}}\right) y=0.75 \mathrm{oz} $$

Graph. Label the solution region.a $$ \begin{aligned} &3 x+2 y<18 \\ &y>4 x+5 \end{aligned} $$
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