Problem 17 - Elementary Algebra

Question

Solve each system by using either the substitution or the elimination-by- addition method, whichever seems more appropriate. $$\left(\begin{array}{l}5 x+3 y=-7 \\ 7 x-3 y=55\end{array}\right)$$

Step-by-Step Solution

Answer

The solution is \(x = 4\), \(y = -9\).

Step 1: Identify the Method

We have two equations: \(5x + 3y = -7\) and \(7x - 3y = 55\). Since the coefficients of \(y\) are opposites (\(+3\) and \(-3\)), using the elimination-by-addition method will simplify the process by directly eliminating \(y\).

Step 2: Eliminate \(y\)

Add the two equations together to eliminate \(y\):\[(5x + 3y) + (7x - 3y) = -7 + 55\]This simplifies to:\[12x = 48\]

Step 3: Solve for \(x\)

Divide both sides of the equation \(12x = 48\) by 12 to solve for \(x\):\[x = \,4\]

Step 4: Substitute \(x\) into the First Equation

Substitute \(x = 4\) back into the first equation \(5x + 3y = -7\):\[5(4) + 3y = -7\]This simplifies to:\[20 + 3y = -7\]

Step 5: Solve for \(y\)

Subtract 20 from both sides of the equation to solve for \(y\):\[3y = -7 - 20\]Simplifying gives:\[3y = -27\]Divide both sides by 3:\[y = -9\]

Key Concepts

Elimination Method

The elimination method is a fantastic technique for solving a system of linear equations. This method is especially useful when the coefficients of one of the variables are opposites. This scenario allows for straightforward elimination of that variable by adding or subtracting the equations. Here is how it works in simple steps:

• Align the equations so that like terms are vertically aligned.
• Add or subtract the equations to eliminate one variable.
• Solve the resulting equation for the remaining variable.
• Substitute the value found back into one of the original equations.
• Solve for the other variable.

This method is clear and efficient. In our case, the equations were perfectly set up for elimination right away, since the coefficients of \(y\) were opposites: \(3\) and \(-3\). Adding the two equations immediately removed \(y\), leaving a simple equation to solve for \(x\). After finding \(x\), we easily substituted back to find \(y\). This gives a complete solution.

Substitution Method

The substitution method is another approach for solving a system of linear equations. This might be chosen when it’s easy to solve for one variable directly from either equation. The main idea here is substituting one variable with an expression found from the other equation.

• Solve. Start by solving one of the equations for one of the variables.
• Substitute. Replace that variable in the other equation with the expression found.
• Simplify. Solve the resulting single-variable equation.
• Back-substitute. Plug the found value back into the expression to find the other variable.

In cases where neither variable easily cancels out, substitution can often be a better option. Although the elimination method was more straightforward for our original exercise, substitution is a robust tool. It allows solving the system step by step through substitution and substitution-back for the unknowns.

Solving Algebraic Equations

Solving algebraic equations is a key skill in mathematics. This skill involves finding the values of variables that satisfy the equation. Both the elimination and substitution methods are forms of solving systems of these equations.

• Intricacy: Algebraic solutions require you to understand variable manipulation techniques.
• Focus: Ensure precision in operations like addition, subtraction, multiplication, and division during the process.
• Checking: After obtaining a solution, plug the values back into both original equations to verify accuracy.

In our given problem, solving involved elemental algebraic operations followed by direct substitution of values. By confirming that \(x = 4\) and \(y = -9\) satisfy both of the original equations, it ensured accuracy in the completed system solution. This acts as a helpful check after calculations.