Problem 25 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{array}{c} {m+2 n=1} \\ {5 m+3 n=-23} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \( m = -19/3, n = 13/3 \)

Step 1: Isolate one variable in the first equation

First, assign \( m = 1 - 2n \) from the first equation. This expression will be used to substitute into the second equation in the next step.

Step 2: Substitute the isolated variable into the second equation and solve

Substitute \( m = 1 - 2n \) into the second equation. This gives us: \(5(1 - 2n) + 3n = -23\). This simplifies to \(3n - 10 = -23\). Solving for \( n \) yields \( n = 13/3 \)

Step 3: Substitute the value of n into the first equation and solve for m

Substitute \( n = 13/3 \) into the first equation to find \( m \). This gives us the equation \(m + 2(13/3) = 1\), simplifying and solving for \( m \) gives \( m = -19/3 \)

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra where the goal is to find the values of unknown variables that make the equation true. When dealing with a simple equation involving only one variable, you typically perform operations to 'isolate' the variable on one side of the equation, leaving the solution on the other. For example, if you have an equation like \( x + 3 = 7 \), you would subtract 3 from both sides to isolate \( x \) and find that \( x = 4 \).

However, when the equation forms a part of a system where two or more equations are interconnected, isolation becomes a strategy within more complex methods, such as substitution, elimination, or matrix operations, to find a solution that satisfies all equations simultaneously.

Algebraic Substitution

Algebraic substitution is a technique used to solve systems of equations by replacing one variable with another expression involving the second variable. This approach simplifies the system to a single variable, making it easier to solve. The key step is to isolate one of the variables in one equation and then 'substitute' this expression into the other equation.

When correctly applied, substitution can unravel systems that may appear complex, transforming them into simpler, solvable equations. This method is highly effective when equations are structured in a way that makes isolation of a variable straightforward, which is often the case with linear systems.

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. In the context of linear systems, these equations represent lines on a coordinate plane, and the solutions represent the points where these lines intersect. Systems can be solved in several ways, with the substitution and elimination methods being the most common for linear systems.

Interpretation in Real Life: Imagine you’re at a market and you want to know the price of apples and bananas. The stall displays two offers: a combination of fruits for certain amounts. By considering these as two equations with the price of apples and bananas as variables, solving the system tells you the price of each fruit separately.

Isolation of Variables

The isolation of variables is the process of rearranging an equation so that one variable stands alone on one side of the equation. It is often the first step in solving both single-variable and multi-variable problems. By manipulating the equation through the inverse operations, such as addition/subtraction or multiplication/division, we effectively 'free' a variable from its coefficients and constants.

Practical Tips: Always work to simplify the equation first, combining like terms and eliminating fractions if possible. Remember to perform the same operation on both sides of the equation to maintain its balance. This is crucial because the equality must hold true throughout the process of solving the equation.