Problem 3 - Algebra 1

Question

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &x+y=300\\\ &x+3 y=18 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of linear equations is x = 441 and y = -141.

Step 1: Arrange the equations

Arrange the system vertically, aligning like terms:\[\begin{align*}x + y &= 300 \\x + 3y &= 18\end{align*}\]

Step 2: Applying the Elimination Method

Subtract the second equation from the first (since the coefficients of 'x' are the same). This will eliminate 'x' and you will get an equation in 'y': \[\begin{align*}(x + y) - (x + 3y) &= 300 - 18 \\-2y &= 282\end{align*}\]

Step 3: Solving for 'y'

Next, divide by -2 to solve for y:\[\begin{align*}y &= \frac{282}{-2} \\y &= -141\end{align*}\]

Step 4: Solving for 'x'

After finding y, substitute y = -141 into the first equation to solve for x:\[\begin{align*}x + y &= 300 \\x - 141 &= 300 \\x &= 300 + 141 \\x &= 441\end{align*}\]

Key Concepts

Elimination Method

The elimination method is a popular technique for solving simultaneous equations. It involves manipulating the equations to eliminate one of the variables, making it easier to solve for the remaining one. This method is especially useful when the coefficients of one of the variables are equal or can easily be made equal.

• Start by aligning the equations so that like terms are vertically aligned.
• Next, look for opportunities to eliminate one variable by adding or subtracting the equations.
• Once a variable is eliminated, solve the resulting single-variable equation.

In the original exercise, the coefficients of 'x' were equal, so by subtracting the second equation from the first, 'x' was eliminated immediately, leaving a simple equation in terms of 'y' to solve. This simplifies the solving process significantly, saving time and reducing potential errors.

Solving Simultaneous Equations

Solving simultaneous equations refers to finding values for the variables that satisfy all given equations simultaneously. These equations can be linear or non-linear, but in this context, we are only dealing with linear equations.

To solve linear simultaneous equations, you have several methods available including substitution, the elimination method, and graphing. Here's a quick overview:

• **Substitution:** Solve one of the equations for one variable and substitute that expression into the other equation.
• **Elimination:** As explained earlier, this involves adding or subtracting equations to eliminate one of the variables.
• **Graphing:** Although not ideal for precision, it can provide a visual approximation of the solution.

For the given problem, the elimination method was chosen because the equation coefficients easily allow one variable to cancel out, streamlining the solution process.

Mathematical Explanation

A mathematical explanation bridges the gap between abstract equations and their practical solutions. It involves understanding the 'why' and 'how' behind each operation in the process of solving equations.

Let's break it down further: - **Why Eliminate?** Choosing to eliminate a variable helps in simplifying the system of equations by reducing two unknowns to one. This isolates the variable we want to solve for. - **Why Subtract?** Subtraction was chosen here because it directly removed one variable, making the path to solving it more straightforward. - **Why Substitute?** After determining one variable through elimination, substitution helps in finding the unknown value of the other variable.
In the provided exercise, these steps led to solving for 'y' once 'x' was successfully eliminated. Then, by back-substitution into one of the original equations, the value of 'x' was found. Each step builds on the previous, ensuring that every calculation reinforces the accuracy of the solution.