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Chapter 5: Problem 91

When solving a system of linear equations in two variables using the substitution or addition method, explain how you can detect whether the system is inconsistent.

Short Answer

Expert verified
A system is inconsistent if the subtraction of the equations results in a false statement (e.g. 0 = 5).

Step by step solution

01

Recognize the System of Linear Equations

Identify the system of linear equations. Let's use the generic example: Equation 1: \[a_1x + b_1y = c_1\]Equation 2: \[a_2x + b_2y = c_2\]
02

Choose the Method - Substitution or Addition

Decide whether to use the substitution method or the addition (elimination) method to solve the system. For now, let's choose the addition method.
03

Multiply to Match Coefficients (Addition Method)

If necessary, multiply one or both equations by a constant to obtain matching coefficients for one of the variables. For example, if the goal is to eliminate \(x\) and our system is:\[ \begin{aligned} \3x + 2y &= 5 \6x + 4y &= 10 \end{aligned} \]We can multiply the first equation by 2 to match the coefficients of \(x\): \[ \begin{aligned} \6x + 4y &= 10 \6x + 4y &= 10 \end{aligned} \]
04

Subtract Equations

Subtract one equation from the other. Using our example:\[ \begin{aligned} \6x + 4y - (6x + 4y) &= 10 - 10 \0 &= 0 \end{aligned} \] If subtraction results in a true statement (like 0 = 0 with both variables eliminated), the system has infinitely many solutions. If it results in a false statement (like 0 = 5), the system is inconsistent.
05

Identify Inconsistency

If subtraction leads to a contradiction, like 0 = 5, it indicates that the system of equations is inconsistent and has no solutions.

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Key Concepts

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

linear equations
Linear equations form the basis of many algebra problems. A typical linear equation looks like this: \(a_1x + b_1y = c_1\), where \(a_1\), \(b_1\), and \(c_1\) are constants, and \(x\) and \(y\) are variables. When you have a system of two linear equations, you can solve them simultaneously to find the values of \(x\) and \(y\). Linear equations can be used in various real-life applications, such as calculating distances, predicting profits, and solving puzzles.
substitution method
The substitution method is one way to solve a system of linear equations. In this approach, you solve one of the equations for one variable and then substitute that expression into the other equation. For example, if you have \(x + y = 5\) and \(2x - y = 3\), you can solve the first equation for \(y = 5 - x\). Then, substitute \(5 - x\) for \(y\) in the second equation to get \(2x - (5 - x) = 3\). Simplify and solve for \(x\). After finding \(x\), substitute it back to find \(y\). This method is efficient if one of the equations can be easily solved for one variable.
addition method
The addition method, also known as the elimination method, involves adding or subtracting the equations to eliminate one of the variables. Suppose you have \(3x + 2y = 5\) and \(6x + 4y = 10\). You can multiply the first equation by 2 to get \(6x + 4y = 10\), which matches the second equation. Subtracting these equations, \(6x + 4y - (6x + 4y) = 10 - 10\), gives \(0 = 0\), which is a true statement indicating infinite solutions. If you get a false statement like \(0 = 5\), the system is inconsistent.
infinite solutions
When a system of linear equations has infinite solutions, it means the equations describe the same line. For instance, if you end up with \(0 = 0\) after elimination, this indicates that the two equations are proportional and represent the same line. In geometric terms, each point on the line is a solution to the system, hence there are infinite solutions. This can be detected by simplifying both equations to see if they are identical.
contradiction in equations
A contradiction in equations occurs when the process of solving a system leads to a false statement, such as \(0 = 5\). This means the system is inconsistent and there are no solutions. Such a contradiction indicates that the two lines represented by the equations are parallel and never intersect. To determine this, eliminate one of the variables and observe the resulting statement. If it's false, you've detected an inconsistency.

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Most popular questions from this chapter

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