Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use any method to solve the system. \(\left\\{\begin{array}{r}-x+3 y=17 \\ 4 x+3 y=7\end{array}\right.\)

Step-by-Step Solution

Answer

The solution for the given system of equations is \( x = 10/3 \) and \( y = 61/9 \).

Step 1: Arrange the System of Equations

The given system of linear equations is: \[\begin{align*} -x + 3y & = 17 \ 4x + 3y & = 7 \end{align*}\] These equations are already arranged in standard form, with the 'x' and 'y' variable terms and constant terms aligned.

Step 2: Apply the Elimination Method

In elimination method, the two equations are added to eliminate one variable. Here, if we add the two equations, the variable 'y' is eliminated. Which yields: \[\begin{align*} -x + 4x & = 17 - 7 \ 3x & = 10 \end{align*}\] The last step yields the equation in one variable 'x'.

Step 3: Solve for 'x'

Now we solve for 'x': \[\begin{align*} x & = 10 / 3 \ x & = 10/3 \end{align*}\]

Step 4: Substitute 'x' in the first equation

Substitute the obtained value of 'x' in the first equation: \[\begin{align*} -10/3 + 3y & = 17 \ 3y & = 17 + 10/3 \ 3y & = 61/3 \ y & = 61/9 \end{align*}\] The above steps result in the solutions for the variables 'x' and 'y'.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. It focuses on eliminating one variable to make the problem easier to solve. Think of it like balancing a scale. If one side is too heavy, you can take away or add weights to balance it out.

Here's how it works in this exercise:

• You have two equations that need solving together.
• By adding or subtracting the equations, you eliminate one of the variables.

In our example, by adding the two equations, the 'y' terms cancel each other out, allowing us to solve for 'x' directly.

Standard Form

Standard form is the format where equations are arranged with variables and constants aligned. This alignment makes it easier to compare and manipulate equations.

The general format is:

• Ax + By = C

Each equation follows this format, making them ready for methods like elimination. The exercise already presents the equations in standard form, highlighting the importance of organized data. This organization helps in visualizing and executing the steps more efficiently.

Variable Elimination

Variable elimination is the goal of the elimination method. By removing one variable, you simplify the equations into a single-variable problem. In this exercise, the 'y' variable is eliminated:

• Add the equations to cancel out the 'y' terms.
• This step reduces the problem to solving a single equation in 'x'.

By focusing on one variable at a time, you streamline the process, making it logical and methodical. It's like peeling an onion layer by layer to get to the core.

Solving for Variables

Once a single variable is isolated, solving it becomes straightforward. For 'x', we ended up with the equation:

• \(3x = 10\)

To solve for 'x':

• Divide both sides by 3, giving \(x = \frac{10}{3}\).

With 'x' known, you substitute back into one of the original equations to find 'y'. This substitution involves basic algebra, and it's the final step to reach a full solution. Each variable holds a piece of the overall answer, completing the puzzle of equations.