Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}3 b+3 m=7 \\ 3 b+5 m=3\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \( m = -2 \) and \( b = \frac{13}{3} \)

Step 1: Set Up the Equations

Formulate the system of equations: \[3b + 3m = 7\] \[3b + 5m = 3\]

Step 2: Eliminate Variable \( b \)

To eliminate \( b \), subtract the second equation from the first one: \[(3b+3m) - (3b+5m) = 7 - 3\] This results in \[-2m = 4\].

Step 3: Solve for \( m \)

To find \( m \), divide both sides of \(-2m = 4\) by -2. You will get \( m = -2 \).

Step 4: Substitute \( m \) into the Original Equations

Now substitute \( m = -2 \) into both of the original equations, for example the first one, check if they hold true. \[3b + 3(-2) = 7 \ 3b - 6 = 7 \ 3b = 13 \ b = \frac {13}{3}\]. Conduct the same check for the second equation.

Step 5: Check Consistency

After we substitute \( m=-2 \) and \( b = \frac {13}{3} \) into both equations, if both equations hold true, then the system is consistent. If one or both equations do not hold true, the system is inconsistent.

Step 6: State the Results

State the solution as \( m = -2, b = \frac{13}{3} \)

Key Concepts

Elimination Method

The elimination method is a popular strategy to solve systems of linear equations. This method involves manipulating the equations in such a way that allows you to eliminate one of the variables.

• The main goal of the elimination method is to cancel out a variable by adding or subtracting the equations.
• Doing this results in a new equation with only one variable, making it easier to solve.
• Once one variable is eliminated, you can solve for the remaining variable and use it to find the other variable.

In our exercise, the aim was to eliminate the variable \( b \) from the system of equations. By subtracting the second equation from the first, \( 3b \) was successfully removed, leaving an equation with only \( m \). This action simplifies the process, paving the way to solving for \( m \) first, then substituting back to find \( b \). The elimination method is efficient and particularly useful when the coefficients of one of the variables are the same or can be easily manipulated to be the same.

Consistent System

A consistent system of equations is one where there is at least one set of values that satisfies all the equations simultaneously. It essentially means that there is a solution intersection between the equations when they are graphed.

• If a system has at least one solution, it is consistent.
• A consistent system can either have exactly one solution or infinitely many solutions.
• In contrast, an inconsistent system of equations has no solutions, indicating that the equations represent parallel lines which never intersect.

For the given set of equations in the exercise, after applying the elimination method and solving for \( m \) and \( b \), it was verified that both equations are satisfied with these values. This confirms the system is consistent, as there exists a unique solution \( (m, b) \) that satisfies both equations.

Solving Equations

Solving equations involves finding the values of the unknown variables that make the equation true. This can involve several methods and steps depending on the complexity of the equation.

• Start by simplifying the equation whenever possible. This could mean combining like terms or clearing decimals and fractions.
• If dealing with a system of equations, decide on the best method: substitution, elimination, or graphing.
• Use algebraic manipulation to isolate the variable you are solving for.
• Substitute the found value back into the original equations to ensure that they hold true.

In our example, the equations were solved by eliminating one variable and then solving the simplified equation. Once \( m \) was found, it was used to calculate \( b \) by substituting back into the original equations.

Linear Equations

Linear equations represent a straight line when graphed on a coordinate plane. These equations are characterized by their standard form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

• They can have one, two, or more variables but always have the degree of 1, which means the variable is not raised to any power other than 1.
• Solving systems of linear equations can provide the point(s) at which the equations intersect, illustrating whether a solution exists.
• Each linear equation can be modified by operations such as addition, subtraction, and multiplication without altering the solution set, which is crucial when using methods like elimination.

The given exercise deals with two linear equations that when solved, provided a specific solution set for the values of \( b \) and \( m \). These solutions confirm that a point of intersection exists for the lines represented by the equations, indicating a consistent system.