Problem 8 - College Algebra Essentials

Question

In Exercises \(5-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} 2 x-3 y=-13 \\ y=2 x+7 \end{array}\right. $$

Step-by-Step Solution

Answer

So, the solution to the system of equations is \(x = -2\) and \(y = 3\).

Step 1: Substitute \(y\) from the second equation into the first one

Substituting the value of \(y\) from the second equation into the first equation gives: \(2x - 3(2x + 7) = -13\).

Step 2: Simplify the resulting equation

Simplifying the above equation gives: \(2x - 6x - 21 = -13\), which simplifies further to: \(-4x - 21 = -13\). Now, add 21 to each side: \(-4x = 8\). Then, divide each side by -4: \(x = -2\).

Step 3: Substitute \(x\) into the second equation to solve for \(y\)

Once we have \(x = -2\), we plug it into the second equation to find \(y\): \(y = 2(-2) + 7 = 3\).

Key Concepts

Substitution Method

The substitution method is a popular technique for solving systems of equations. It involves substituting one variable in one equation with its expression obtained from another equation. This way, a system of two variables is transformed into a single-variable equation, making it easier to solve.
For the given system of equations, you start with two linear equations:

• Equation 1: \(2x - 3y = -13\)
• Equation 2: \(y = 2x + 7\)

Since Equation 2 already expresses \(y\) in terms of \(x\), we use this expression and substitute \(y\) in Equation 1. This substitution method simplifies the problem, allowing you to isolate the variable \(x\) and solve for it directly. After finding \(x\), it’s easy to substitute back to find \(y\). This method is especially useful when one of the equations is already solved for one variable.

Linear Equations

Linear equations are mathematical expressions involving constants and variables raised only to the first power. They have straightforward solutions because they form straight lines when graphed on a coordinate plane.
Each equation in the presented system is a linear equation. The goal here is to find the point of intersection of these lines, which represents the solution to the system of equations. For this system:

• First equation: \(2x - 3y = -13\)
• Second equation: \(y = 2x + 7\)

Both equations describe lines in a two-dimensional space. Solving them means finding the values of \(x\) and \(y\) where these lines intersect. In simple terms, these values satisfy both equations simultaneously.

Algebraic Solutions

Algebraic solutions involve using algebraic techniques to solve equations. The primary tools include substitution, simplification, and sometimes factoring.
In solving our system with the substitution method, we first used algebra to substitute \(y\) in the first equation, converting it into a single equation in terms of \(x\). We performed the following steps:

• Simplification: Transform \(2x - 3(2x + 7) = -13\) into \(-4x - 21 = -13\)
• Isolate \(x\): Solve \(-4x = 8\) to find \(x = -2\)
• Plugging back: Use \(x = -2\) to find \(y\) from \(y = 2x + 7\)

These methods help derive the solutions systematically and ensure you find accurate results. Using algebraic solutions provides a clear, step-by-step framework that helps develop problem-solving skills across various math applications.