Problem 67 - College Algebra Essentials

Question

Solve by the addition method: $$ \left\\{\begin{array}{l} 2 x+4 y=-4 \\ 3 x+5 y=-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 4\), \(y = -3\)

Step 1: Identify the system of equations

Our system of equations given is \(2x + 4y = -4\) and \(3x + 5y = -3\)

Step 2: Align the coefficients of the variable to be eliminated

We can multiply the first equation by 3 and the second equation by 2, this gives us \(6x + 12y = -12\) and \(6x + 10y = -6\)

Step 3: Eliminate one variable by subtracting the second equation from the first

Subtracting gives \(2y = -6\), which simplifies to \(y = -6/2 = -3\)

Step 4: Substitute y in the first equation

Substituting \(y = -3\) into the original first equation \(2x + 4y = -4\) gives \(2x + 4(-3) = -4\), which simplifies to \(2x = -4 +12\), further simplification gives \(x = 4\)

Step 5: Interpret the solution

So the solution to the system of equations is \(x = 4\) and \(y = -3\)

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a powerful tool for solving systems of linear equations. It involves manipulating the equations so that when you add or subtract them, one of the variables is eliminated. This simplifies the system, making it easier to solve.

• Choose a variable to eliminate: In our example, the system of equations is \(2x + 4y = -4\) and \(3x + 5y = -3\). We aim to eliminate either \(x\) or \(y\).
• Adjust coefficients: Multiply each equation by a suitable number so that the coefficients of the chosen variable are the same. For instance, we multiplied the first equation by 3, and the second by 2 to align the coefficients of \(x\).
• Eliminate the variable: Add or subtract the equations. In our case, subtracting eliminates the \(x\)-variable, leaving an equation in terms of \(y\) only.

By simplifying one equation, we pave the way for an easier substitution process to find the value of the other variable.

Solution of Linear Equations

Solving linear equations is a fundamental skill, focusing on finding the values of variables that satisfy all equations in a system. In this exercise, our system consists of two linear equations, each with two variables. Our goal is to find a common solution for \(x\) and \(y\) that makes both equations true simultaneously.

• Simplify and solve for one variable: After eliminating \(x\), we were left with \(2y = -6\). Solving gives \(y = -3\).
• Substitute back to find the second variable: Once \(y\) is found, substitute it into one of the original equations to solve for \(x\). Here, substituting \(y = -3\) into \(2x + 4(-3) = -4\) allowed us to solve for \(x\).
• Check your solution: Always substitute your solutions back into the original equations to verify their correctness. In this case, \(x = 4, y = -3\) is checked against both starting equations.

Understanding the procedure for solving linear equations helps improve problem-solving skills, applicable to various mathematical contexts.

Elimination of Variables

The elimination of variables is a technique used to simplify systems of equations by removing one variable, thus reducing the system to a single equation. This technique often involves addition or subtraction strategies.

• Align variables: As seen, to eliminate \(x\), the coefficients were manipulated so both equations had the same \(x\) coefficient.
• Choice of operation: After aligning, subtraction was used to remove \(x\), leaving a simpler equation for \(y\).
• Solve the simplified equation: With \(2y = -6\) derived, determining \(y\) was straightforward.

Eliminating variables effectively turns complex systems into simpler linear equations. This method provides clarity and precision in finding solutions and is an essential component of algebraic problem-solving.