Problem 15 - Algebra 1

Question

Tell which equation you would use to isolate a variable. Explain your reasoning. $$\begin{aligned} &m+4 n=30\\\ &m-2 n=0 \end{aligned}$$

Step-by-Step Solution

Answer

The second equation \( m-2n=0 \) should be used to isolate a variable. Through manipulation, it gives \( m = 2n \), this expression can then be substituted into the first equation, leading to resolutions for both variables \( m = 10 \) and \( n = 5 \)

Step 1: Isolate the desired variable

The second equation is chosen because it has a simpler form. Therefore, it's easier to choose any variable between \( m \) and \( n \) to isolate. Let's isolate \( m \) from second equation, which gives: \( m = 2n \).

Step 2: Substitution

Now that \( m \) has been expressed in terms of \( n \), this expression can be substituted into the first equation. This gives a new equation: \( 2n + 4n = 30 \).

Step 3: Solve for the Other Variable

Resolve the equation from step 2 to solve for \( n \), the result is \( n = 5 \)

Step 4: Solve for Isolated Variable

Substitute the value of \( n \) into the equation \( m = 2n \) to find \( m \). The result is \( m = 10 \).

Key Concepts

Substitution Method

The substitution method is a fundamental technique for solving systems of equations. This method allows you to solve two or more equations by expressing one variable in terms of another. In this approach, you start by rearranging one of the equations to isolate one of the variables. Once you have isolated the variable, it provides a way to substitute this expression into another equation.

Consider the system of equations given in the exercise:

• Equation 1: \( m + 4n = 30 \)
• Equation 2: \( m - 2n = 0 \)

For the substitution method, step 1 focuses on isolating a variable in one of these equations. Once this is achieved, as we see in the step-by-step solution, the chosen equation is the simpler one, \( m - 2n = 0 \). This choice simplifies the process of substitution because it allows us to express \( m \) in terms of \( n \). By substituting \( m = 2n \) from equation 2 into equation 1, we form a new equation solely in terms of \( n \). This method simplifies a system of equations to a single equation with one variable, making it easier to solve.

Isolating Variables

Isolating variables is a crucial step in solving any equation and is particularly useful when applying the substitution method. When you isolate a variable, you rearrange the equation so that the variable you are solving for is by itself on one side of the equation. This helps to simplify complex systems of equations and make substitution possible.

In the exercise, isolating a variable occurs in the second equation, \( m - 2n = 0 \). This equation is selected because it's straightforward to manipulate, allowing us to isolate \( m \) easily. Here's how it's done:

• Identify the variable to isolate. In our case, \( m \).
• Rearrange the equation so that the chosen variable is on one side. For example, adding \( 2n \) to both sides gives: \( m = 2n \).

Once isolated, \( m \) in terms of \( n \) (\( m = 2n \)) can be utilized effectively in the first equation to solve for \( n \). Isolating variables is often the first move when breaking down the process of substitution or elimination and is vital for finding solutions to systems of equations.

Linear Equations

Linear equations form the backbone of algebra and are essential in solving systems of equations. A linear equation is any equation that can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. The equations given in the original exercise are both linear:

• \( m + 4n = 30 \)
• \( m - 2n = 0 \)

Linear equations have specific properties that make them easier to work with:

• They graph as straight lines on a coordinate plane.
• Each solution of a linear equation corresponds to a point on the line.
• They can be added, subtracted, and manipulated algebraically.

Understanding these properties is important because it guides how we can manipulate and solve them. Each variable and its coefficient in a linear equation can be isolated or expressed in a simpler form, allowing for techniques like substitution to be applied smoothly. When solving these equations, you often aim to find the values of \( m \) and \( n \) that satisfy both equations simultaneously, signifying the intersection of their corresponding lines on a graph.