Problem 10 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(m+3 n=2\) \(-m+2 n=3\)

Step-by-Step Solution

Answer

After solving, we find that the solution is \(m = -1\) and \(n = 1\).

Step 1: Add the two equations

We first add the two equations to find a new equation. Adding the left sides of the two equations: \(m + 3n + -m + 2n\) and the right sides \(2 + 3\) of the equations we get \(5n = 5\).

Step 2: Solve for n

Next, we isolate \(n\) by dividing both sides of the equation by 5, which yields \(n = 1\).

Step 3: Substitute n=1 in the first equation

Once \(n\) is found, we substitute \(n = 1\) into the first equation: \(m + 3*1 = 2\), which simplifies to \(m = -1\).

Step 4: Check the solution

Finally, we verify these solutions by substituting \(m = -1\) and \(n = 1\) into the second equation. The left hand side becomes \(-(-1) + 2*1 = 3\), which equals to the right hand side, confirming that the solution is accurate.

Key Concepts

Linear System

A linear system is a collection of two or more linear equations involving the same set of variables. In this case, we have two equations with two unknowns, represented as:

• \( m + 3n = 2 \)
• \( -m + 2n = 3 \)

These equations are linear because they graph as straight lines on a coordinate plane. The objective of solving a linear system is to find the values of the variables that satisfy all equations concurrently. Solving these systems helps us understand the relationship between the variables and how one variable changes with respect to the other.

Solution Verification

Solution verification is an indispensable step in solving a system of equations. After obtaining a potential solution, it's crucial to verify it satisfies all original equations. This confirms the accuracy of the solutions. In our case:

• We found \( n = 1 \) and \( m = -1 \).
• Insert these values back into the original equations.
• For \( m + 3n = 2 \): Substitute \( m = -1 \) and \( n = 1 \). It simplifies to \( -1 + 3*1 = 2 \), which is true.
• For \( -m + 2n = 3 \): Substitute \( m = -1 \) and \( n = 1 \). It simplifies to \( -(-1) + 2*1 = 3 \), also true.

This step confirms that our solutions are correct and complete.

Substitution Method

The substitution method is one strategy to find solutions of a linear system. It involves expressing one variable from one of the equations and substituting it in the other. In this exercise:

• First, solve one equation for one variable. We did this indirectly in the step-by-step solution by adding and eliminating \( m \).
• Once \( n = 1 \) was found, substitute this value back into an equation to solve for \( m \).
• In a more typical approach, you would express \( m \) in terms of \( n \) or vice versa directly and substitute back.

This method makes it easy to isolate and focus on solving for one variable at a time.

Equation Solving

Equation solving involves finding specific values for variables that satisfy a given equation or set of equations. Often this requires manipulating the equations algebraically:

• Combine terms by addition or subtraction.
• Isolate terms with the variable of interest by using multiplication or division.
• Simplify equations to deduce clear solutions.

For this set of equations, adding them simplified the problem by eliminating \( m \) altogether, thus isolating \( n \). Once \( n \) was known, substituting it back into one of the original equations allowed us to find \( m \). This process highlights the importance of using strategic moves at each step to solve the equations efficiently.