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Chapter 4: Problem 31

Solve each system. $$\begin{aligned}x+y+z &=9 \\\x+y &=5 \\\z &=1\end{aligned}$$

Short Answer

Expert verified
There is no solution to the system due to inconsistency.

Step by step solution

01

Substitute the third equation into the first

The third equation is given as \(z=1\). Substitute \(z=1\) into the first equation \(x + y + z = 9\). This gives \[x + y + 1 = 9\].
02

Simplify the equation

Simplify the equation from Step 1: \(x + y + 1 = 9\). Subtract 1 from both sides to get \(x + y = 8\).
03

Compare with the second equation

Notice that the equation from Step 2 \(x + y = 8\) should be compared with the second given equation \(x + y = 5\). Since these two equations are contradictory, there is an inconsistency.

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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 popular technique for solving systems of equations. It involves substituting one equation into another, to reduce the number of variables, making it easier to solve the system. Let's break down how we applied the substitution method in the given exercise.
First, we identify an equation that can easily be solved for one of the variables. In this case, our third equation is already neatly solved for the variable \(z\):
  • \(z = 1\)
We then substitute this value of \(z\) into the first equation:
  • Substitute \(z=1\)
  • This turns the first equation into \(x + y + 1 = 9\)
Thus, with one variable out of the way through substitution, we simplify the equation later on, reducing its complexity.
Inconsistent Systems
An inconsistent system of equations has no solution. This occurs when the equations contradict each other, meaning there are no values for the variables that can satisfy all the equations simultaneously.
In our exercise, after we substituted and simplified, we compared the equations:
  • \(x + y = 8\) (from substitution)
  • \(x + y = 5\) (second given equation)
At this point, we noticed a contradiction. The equation \(x + y\) cannot equal both 8 and 5 at the same time.
This clear discrepancy tells us that the system is inconsistent, and therefore, it has no solution.
Simplification
Simplification in solving systems of equations involves reducing an equation to its simplest form to make it easier to solve. Let鈥檚 see how simplification worked in our exercise after applying the substitution method:
After substituting \(z\) from the third equation into the first, we got:
  • \(x + y + 1 = 9\)
We simplified this by subtracting 1 from both sides:
  • \(x + y = 8\)
This step was crucial because it helped us directly compare this simplified equation to the second given equation.
Simplification is an important step in solving equations as it reveals inconsistencies and contradictions more clearly, as evidenced in the comparison of the simplified equations \(x + y = 8\) and \(x + y = 5\).

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Most popular questions from this chapter

Write a system of two equations in two unknowns for each problem. Solve each system by substitution. Investing her bonus. Donna invested her \(\$ 33,000\) bonus and received a total of \(\$ 970\) in interest after one year. If part of the money returned \(4 \%\) and the remainder \(2.25 \%\) then how much did she invest at each rate?

Evaluate the determinant of each \(3 \times 3\) matrix using expansion by minors about the row or column of your choice. \(\left[\begin{array}{rrr}-2 & -3 & 0 \\ 4 & -1 & 0 \\ 0 & 3 & 5\end{array}\right]\)

Solve each problem using a system of linear equations and the Gauss-Jordan elimination method. Two more numbers. The sum of two numbers is 11 and their difference is \(6 .\) Find the numbers.

Determine the row operation that was used to convert each given augmented matrix into the equivalent augmented matrix that follows it. $$ \left[\begin{array}{rr|r} 3 & 2 & 12 \\ 1 & -1 & -1 \end{array}\right],\left[\begin{array}{rr|r} 1 & -1 & -1 \\ 3 & 2 & 12 \end{array}\right] $$

Determine the size of each matrix. $$ \left[\begin{array}{lll} 2 & -3 & 5 \end{array}\right] $$
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