Problem 14 - Algebra 1

Question

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &2 x+3 y=3\\\ &5 x+5 y=10 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 3\) and \(y = -1\). The elimination method was chosen because the equations in the system were simplified, making the elimination process straightforward.

Step 1: Simplify the Equations

Start by simplifying the equations as much as possible. This often involves dividing through by common factors. For the second equation,\(5x + 5y = 10\), divide each term by 5 to get \(x + y = 2\). Now the system of equations is: \(\begin{align*}&2x + 3y = 3\&x + y = 2\end{align*}\)

Step 2: Elimination Method Preparation

To apply the elimination method, make the coefficient of x or y the same in both equations, then subtract one equation from the other. In this case, multiply the second equation by 2 to make the coefficient of x the same in both equations. This results in the following system:\(\begin{align*}&2x + 3y = 3\&2x + 2y = 4\end{align*}\)

Step 3: Apply the Elimination Method

Subtract the second equation from the first to eliminate x. The calculation goes as follows: \((2x + 3y) - (2x + 2y) = 3 - 4\), which leaves \(y = -1\). Thus we have found the value of y.

Step 4: Solve for x

Substitute y = -1 into the simplified second equation from Step 1 to find the value of x. The calculation goes as follows: \(x + (-1) = 2\), which leaves \(x = 3\). Thus we have found the value of x and solved the whole system.

Key Concepts

Elimination Method

The elimination method is a powerful technique for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variables.
To use this method effectively, it is important to align the equations such that one variable can be easily canceled out.

• Start by modifying the equations so that the coefficients of one variable are the same or additive opposites.
• Once aligned, add or subtract the equations to eliminate that variable, leaving a single-variable equation to solve.
• With the variable eliminated, solve for the remaining variable and back substitute to find the other variable.

Choosing when to use the elimination method largely depends on the equations given.
If one or more equations can be easily manipulated to have matching coefficients, this method tends to be very efficient and quick.

Simplifying Equations

Simplifying equations is a crucial step before applying any method to solve linear systems. It involves reducing equations to simpler forms that are easier to work with.
A common technique is dividing all terms by a common factor, simplifying fractions, or combining like terms.

• The goal is to transform the system into a simpler equivalent system without changing its solutions.
• In the provided example, simplifying was achieved by dividing the entire second equation by 5, allowing for easier manipulation in subsequent steps.
• This process makes other methods, like elimination or substitution, more straightforward and less error-prone.

Simplifying is often the first step taken to minimize the complexity of a linear system, streamlining the entire solving process.

Substitution Method

The substitution method is particularly useful when one of the equations in a system is easy to solve for one variable. After isolating one variable, substitute it back into the other equation to find the second variable.
Here’s how it works:

• Solve one of the equations for one variable in terms of the other(s).
• Substitute this expression into the other equation, transforming it into a single-variable equation.
• Solve this new equation, and use the solution to find the other variable by back-substituting.

The substitution method is ideal for systems where isolation of a variable is straightforward and results in simple calculations.
Although not used in the original solution, knowing how to apply this method provides a flexible alternative to elimination, especially for systems amenable to early isolation of variables.