Problem 10 - Algebra and Trigonometry

Question

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{4 x-3 y=11} \\ {8 x+4 y=12}\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = -1 \).

Step 1: Arrange Equations for Elimination

Start with the system of equations: \[ \begin{cases} 4x - 3y = 11 \ 8x + 4y = 12 \end{cases} \] We need the coefficients of one of the variables to be the same in both equations so they can be eliminated when subtracted.

Step 2: Multiply Equations to Match Coefficients

Multiply the first equation by 2 to make the coefficient of \ x in the first equation the same as in the second. You get: \[ 8x - 6y = 22 \] Now, the set becomes: \[ \begin{cases} 8x - 6y = 22 \ 8x + 4y = 12 \end{cases} \]

Step 3: Subtract to Eliminate Variable

Subtract the second equation from the first to eliminate \ x: \[ (8x - 6y) - (8x + 4y) = 22 - 12 \] Simplifying gives: \[ -10y = 10 \]

Step 4: Solve for y

Solve for \ y by dividing both sides by -10: \[ y = -1 \]

Step 5: Substitute to Find x

Substitute \ y = -1 into the original first equation: \[ 4x - 3(-1) = 11 \] This simplifies to: \[ 4x + 3 = 11 \] Solve for \ x by first subtracting 3 from both sides: \[ 4x = 8 \] Then divide by 4: \[ x = 2 \]

Step 6: Verify the Solution

Substitute \ x = 2 and \ y = -1 into the second original equation for verification: \[ 8(2) + 4(-1) = 16 - 4 = 12 \] Since both sides of the equation are equal, the solution is correct.

Key Concepts

Understanding a System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In our case, we have a system of linear equations involving variables \(x\) and \(y\). The main aim here is to find values for these variables that make all the equations true simultaneously.
Such a system can be written in a bracketed form showing each equation as a separate component. It's typically expressed as:

• \(4x - 3y = 11\)
• \(8x + 4y = 12\)

To solve these, the elimination method is often applied because it's a reliable approach, especially with linear equations. This technique aims to eliminate one of the variables by adding or subtracting the equations, allowing you to solve for the remaining variable first. Once one variable is known, you can easily find the other.
Understanding the structure of systems of equations helps in choosing the right methods to solve them efficiently.

Elimination Method to Solve for Variables

When dealing with a system of equations, solving for variables is the ultimate goal. The elimination method allows you to do this by systematically removing one variable to solve for the other. Let's see how this works in practice.
First, we have the system:

• \(4x - 3y = 11\)
• \(8x + 4y = 12\)

To eliminate \(x\), we manipulate the equations so that the coefficients of \(x\) in both equations become equal. By multiplying the first equation by 2, the x-coefficients match, resulting in:

• \(8x - 6y = 22\)
• \(8x + 4y = 12\)

Next, subtract the second equation from the first to eliminate \(x\):
\(-10y = 10\)
Solving for \(y\) involves dividing both sides by -10, yielding \(y = -1\).
With \(y\) known, substitute it back into one of the original equations to find \(x\). Using \(4x - 3y = 11\), plug in \(y = -1\) to get \(x = 2\).
By using elimination effectively, we are able to isolate and solve for each variable in a systematic way.

Exploring Linear Equations in the Context of Elimination Method

Linear equations are equations of the first order, where each term is either a constant or the product of a constant and a single variable.
They are called "linear" because they graph as straight lines in a two-dimensional space, defined typically by the general form \(ax + by = c\). Each equation in our initial system of equations, \(4x - 3y = 11\) and \(8x + 4y = 12\), follows this structure.
With the use of the elimination method, linear equations like these can be solved quite effectively:

• Align coefficients such that one variable can be easily cancelled out.
• Apply basic arithmetic operations such as multiplication, addition, or subtraction.
• Solve for one variable once the system is reduced to a single linear equation.

Understanding that these equations remain linear throughout the elimination process is key; they merely combine and transform but maintain their linear properties, allowing for straightforward calculation.
This makes solving them methodical and reliable, especially for systems with two variables, ensuring that each step reinforces the ultimate goal of finding solutions for both \(x\) and \(y\).