Problem 58 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. \(m+2 n=1\) \(5 m-4 n=-23\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(m = -3\) and \(n = 2\).

Step 1: Isolate one variable in the first equation

Start by solving the first equation for m: \(m = 1 - 2n\)

Step 2: Substitute into the second equation

Substitute \(m = 1 - 2n\) into the second equation to get a new equation: \(5(1 - 2n) - 4n = -23\). This simplifies to: \(5 - 10n - 4n = -23\), or \(14n = 28\)

Step 3: Solve for n

Solve for n by dividing both sides of the equation by 14, obtaining \(n = 2\)

Step 4: Substitute n back into the first equation

Plug \(n = 2\) back into the first equation \(m = 1 - 2*2\) so that \(m = -3\)

Key Concepts

Understanding Linear Systems

A linear system consists of two or more linear equations with two or more variables. In this instance, our linear system comprises the equations \(m+2n=1\) and \(5m-4n=-23\). These equations involve the variables \(m\) and \(n\), and our task is to find values for these variables that satisfy both equations simultaneously. A linear system is often visualized as two lines on a graph, where the solution corresponds to the point at which the lines intersect. Solving linear systems can be done using various methods such as graphing, substitution, or elimination. The substitution method is particularly useful when one of the equations is easy to manipulate for isolating a variable.

Isolating a Variable

Isolating a variable is a foundational skill in algebra, which involves rearranging an equation so that one variable stands by itself on one side of the equation. For instance, in our exercise, we start with the equation \(m + 2n = 1\). By isolating \(m\), we rearrange it to become \(m = 1 - 2n\). This step is crucial in the substitution method because it sets up the opportunity to replace \(m\) in the second equation with an expression that involves only \(n\). To isolate \(m\), we simply subtracted \(2n\) from both sides of the equation. This process helps to simplify complex linear systems by reducing the number of variables in our work.

Solving for n

After isolating the variable \(m\), we move on to finding the value of \(n\). Once we have substituted \(m = 1 - 2n\) into the second equation \(5m - 4n = -23\), the equation becomes \(5(1 - 2n) - 4n = -23\). This simplifies through distribution to \(5 - 10n - 4n = -23\), or \(-14n = -28\). Solving for \(n\), we divide both sides by \(-14\), obtaining \(n = 2\). Taking it step by step:

• Distribute the 5 in the expanded equation.
• Simplify by combining like terms to gather all \(-n\) terms together.
• Isolate \(n\) by performing the inverse operation of multiplication.

Now we have the value of \(n\), making us one step closer to solving the system.

Plugging Back Into the Equation

With the value of \(n\) found, we can determine \(m\) by plugging \(n = 2\) back into our isolated equation \(m = 1 - 2n\). Doing so, we substitute \(n\) to get \(m = 1 - 2 \times 2 = -3\). This final step allows us to determine the exact pair \((m, n)\) that serves as the solution to the system of equations, confirming that \(m = -3\) and \(n = 2\). When plugging back, make sure:

• To substitute \(n\) into the exact form of the equation you derived originally, to avoid errors.
• Evaluate expression accurately to ensure the right answer.

This demonstrates the power of the substitution method to break down and solve complex linear systems efficiently.