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Chapter 4: Problem 72

When using the addition method, how can you tell if a system of linear equations has no solution?

Short Answer

Expert verified
In the Addition/Elimination method, if all variables are eliminated and results in an untrue statement after the addition or subtraction process, this indicates that the system of equations has no solution.

Step by step solution

01

Understand Addition/Elimination Method

The Addition/Elimination method involves adding or subtracting the equations in order to eliminate one of the variables. This allows you to solve for the remaining variable.
02

Analyze the System of Equations

When using the Addition/Elimination method to solve a system of linear equations, it's possible to end up with an equation that is not possible, such as 0 = 5. This happens when both equations are land up on top of each other after the addition or subtraction process, which is possible when the two lines represented by the equations are parallel.
03

Identify Indication of No Solution

If after adding or subtracting the two equations in the system, you find that all the variables cancel each other out and leave behind an untrue statement (like 0 = 5), this means the system of equations has no solution. This is because you've essentially proved that the two lines never intersect, thus there is no common solution to both equations.

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Most popular questions from this chapter

Verify your solutions to any five exercises from Exercises 11 through 36 by using a graphing utility to graph the two equations in the system in the same viewing rectangle. After entering the two equations, one as \(y_{1}\) and the other as \(y_{2},\) and graphing them, use the \([\text { INTERSECTION }]\) feature to find the system's solution. (It may first be necessary to solve the equations for \(y\) before entering them.)

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} 2(x+y)=4 x+1 \\ 3(x-y)=x+y-3 \end{array}\right.$$

If \(x=3-y-z, 2 x+y-z=-6,\) and $$3 x-y+z=11,\( find the values for \)x, y,\( and \)z$$

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} x-y=-3 \\ \frac{x}{9}-\frac{y}{7}=-1 \end{array}\right.$$

Involve mixtures How many ounces of a \(50 \%\) alcohol solution must be mixed with 80 ounces of a \(20 \%\) alcohol solution to make a \(40 \%\) alcohol solution?
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