Problem 24 - Elementary Algebra

Question

Solve each system by using either the substitution or the elimination-by- addition method, whichever seems more appropriate. $$\left(\begin{array}{l}3 x-y=9 \\ 5 x+7 y=1\end{array}\right)$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{32}{13}\), \(y = -\frac{21}{13}\).

Step 1: Decide the Method

First, we examine the system of equations: \(3x - y = 9\) and \(5x + 7y = 1\). Since the first equation is simpler and easily solved for \(y\), we'll use substitution.

Step 2: Solve for y in the First Equation

Take the first equation \(3x - y = 9\). Solve for \(y\) by adding \(3x\) to both sides and then simplifying: \(y = 3x - 9\).

Step 3: Substitute y in the Second Equation

Replace \(y\) in the second equation \(5x + 7y = 1\) with the expression from Step 2: \(5x + 7(3x - 9) = 1\).

Step 4: Simplify and Solve for x

Distribute and simplify the equation: \(5x + 21x - 63 = 1\). Combine like terms to get: \(26x - 63 = 1\). Add 63 to both sides: \(26x = 64\). Divide both sides by 26 to solve for \(x\): \(x = \frac{64}{26} = \frac{32}{13}\).

Step 5: Solve for y Using x Value

Substitute the value found for \(x\) back into the expression for \(y\): \(y = 3\left(\frac{32}{13}\right) - 9\). Simplify to find \(y = \frac{96}{13} - \frac{117}{13}\). Combine the fractions to get \(y = -\frac{21}{13}\).

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, especially when one of the equations can be easily solved for one variable.
To use substitution, we first solve one of the equations for one variable, such as finding an expression for either \(x\) or \(y\).
Once we have this expression, we substitute it into the other equation. This allows us to transform the system of two equations into a single equation in one variable.
For example, consider the system:

• \(3x - y = 9\)
• \(5x + 7y = 1\)

We can solve the first equation for \(y\):\(y = 3x - 9\).
By substituting \(y = 3x - 9\) into the second equation, we replace \(y\) with the expression, leading to an equation solely in terms of \(x\): \(5x + 7(3x - 9) = 1\).
This simplification turns the problem into one that is more straightforward to solve.

Elimination Method

The elimination method is another algebraic technique that simplifies solving a system by eliminating one variable. This method is usually more appropriate when the coefficients before variables allow easy elimination, often through addition or subtraction.
Here's how elimination works:

• Line up the equations so they are in a similar form, typically with \(x\) and \(y\) terms on one side and constants on the other.
• Multiply one or both equations, if needed, to get opposite coefficients for one variable.
• Add or subtract the equations to eliminate one variable, simplifying the system to a single equation with one unknown.

For the equations \(3x - y = 9\) and \(5x + 7y = 1\), elimination isn't the first choice as neither variable can be easily eliminated without multiplying both equations. However, this method can be helpful when the coefficients are set for easy addition or subtraction.

Algebraic Techniques

In solving systems of equations, algebraic techniques like substitution and elimination are key tools. These methods allow us to handle systems by focusing on either isolating variables or eliminating them altogether.
Algebraic techniques are not just limited to simple substitution and elimination. They often include strategies for simplifying complex expressions, finding least common multiples for easier elimination, or using matrix operations in larger systems.
Some tips for effective use of algebraic techniques:

• Always simplify equations whenever possible, which makes subsequent steps easier.
• Check initial equations for opportunities to easily isolate or eliminate variables.
• Remember to verify your solutions by substituting values back into original equations.

The beauty of algebraic methods is their flexibility. Whether you're using substitution or elimination, keeping these techniques and tips in mind can make solving systems of equations more intuitive and manageable.