Problem 2 - Elementary Algebra

Question

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair. $$ m=-6 n+1, \text { if } n=2 $$

Step-by-Step Solution

Answer

Answer: The value of 'm' when $$n=2$$ is $$m=-11$$.

Step 1: Write Down the Given Equation and Information

The given linear equation is: $$ m = -6n + 1 $$ And we have the information that: $$ n = 2 $$

Step 2: Substitute the Value of 'n' into the Equation

Now, we will substitute the value of $$n$$ (which is 2) into the given equation: $$ m = -6(2) + 1 $$

Step 3: Solve for 'm'

Let's solve the equation for $$m$$: $$ m = -12 + 1 $$ $$ m = -11 $$

Step 4: Write the Solution as an Ordered Pair

Finally, we'll write the solution as an ordered pair (m, n), where $$m=-11$$ and $$n=2$$: $$ (-11, 2) $$ So, the solution to the given linear equation in two variables is $$(-11, 2)$$.

Key Concepts

Ordered Pair

An ordered pair is a fundamental concept in algebra, particularly when dealing with two-dimensional coordinate systems. It's a set of two numbers written in a specific order, usually within parentheses, and separated by a comma, like this: \( (x, y) \). The first number, 'x', represents the horizontal position, while the second number, 'y', indicates the vertical position on a coordinate plane.

Each ordered pair corresponds to a single point on the plane, and no two distinct points can have the same ordered pair of coordinates. When solving linear equations with two variables, the solutions can be expressed as ordered pairs. As in the given exercise, once we find the values of 'm' and 'n', we form the ordered pair \( (-11, 2) \) to represent the solution on a coordinate plane.

Substituting Values

In algebra, substituting values is a critical technique for finding variable specific outcomes within an equation. This process involves replacing a variable with its given or calculated value. For example, in the exercise above, the value of 'n' is given as 2. So we literally place 2 where 'n' appears in the equation, which leads to a simplified expression that can be easily solved. This method can be particularly useful when dealing with equations with multiple variables.

• Identify the variable that you need to substitute.
• Replace that variable with its given numerical value.
• Simplify the equation if necessary.
• Solve for the remaining variable(s).

By consistently applying the substitution method, students can solve equations more efficiently and with greater understanding.

Solving Linear Equations

The process of solving linear equations requires finding the value(s) of the variable(s) that make the equation true. These equations can range from the very simple, like \( x + 2 = 5 \), to more complex equations with multiple variables, such as the one provided in the exercise.

There are several methods to solve linear equations, but one of the most common is through simplification and isolation:

1. Rearrange the equation to isolate the variable on one side of the equation.
2. Simplify the equation by combining like terms and using basic arithmetic operations.
3. Check the solution by substituting it back into the original equation.

This technique is a cornerstone in algebra and serves as a foundation for more advanced mathematical concepts. Remember, the goal is to get the variable on its own, with a coefficient of 1, so its value can be clearly identified.