Problem 12 - Algebra and Trigonometry

Question

\(9-12\) . 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: Set Up the System of Equations

Consider the system of equations given by \( 4x - 3y = 11 \) and \( 8x + 4y = 12 \). We will eliminate one variable to solve the system.

Step 2: Align the Equations for Elimination

To eliminate \( x \), we need to make the coefficients of \( x \) in both equations the same. Multiply the first equation by 2 to make the coefficients of \( x \) equal in both equations.

Step 3: Modify the First Equation

Multiply the first equation by 2: \((2)(4x - 3y) = (2)(11)\). This gives us the new equation: \(8x - 6y = 22\).

Step 4: Subtract the Modified Equations

Subtract the second equation from the new first equation:\[(8x - 6y = 22) \] \[ -(8x + 4y = 12) \] which simplifies to \(-10y = 10\).

Step 5: Solve for \( y \)

Divide both sides of the equation by -10 to solve for \( y \):\(-10y = 10\)\(y = -1\).

Step 6: Substitute \( y \) into Original Equation

Substitute \( y = -1 \) back into one of the original equations, for example, \( 4x - 3y = 11 \):\(4x - 3(-1) = 11\)\(4x + 3 = 11\).

Step 7: Solve for \( x \)

From the equation \(4x + 3 = 11\), subtract 3 from both sides:\(4x = 8\). Divide by 4:\(x = 2\).

Step 8: Final Step: Verify the Solution

Verify by substituting \( x = 2 \) and \( y = -1 \) into both original equations:1. \(4(2) - 3(-1) = 11\) checks out.2. \(8(2) + 4(-1) = 12\) checks out. Therefore, the solution is correct.

Key Concepts

System of Equations

A system of equations consists of two or more equations that we need to solve simultaneously. These equations have common variables. The goal is to find the values of these variables that satisfy all equations at the same time.
In many real-world applications, systems of equations model interactions where different relationships are interconnected.
In this exercise, we have a system with two linear equations:

• Equation 1: \(4x - 3y = 11\)
• Equation 2: \(8x + 4y = 12\)

These represent straight lines in a coordinate plane. Solving the system of equations means finding the point(s) where these two lines intersect.

Solving Linear Equations

Solving a linear equation involves finding the value of the variable that makes the equation true. Linear equations have no variables raised to any power other than one. They are "straightforward," so to speak, due to the direct relationship between the variables.
The core principle of solving these equations involves isolating the variable on one side of the equality. In this exercise, the first step was to transform the system into a form that allows for elimination of a variable. Aligning coefficients in both equations makes it easier to cancel out one variable and solve for the other. This is part of using the elimination method, where you first adjust the coefficients and then perform operations to consolidate the system.

Variable Elimination

Variable elimination is a strategy used to remove one variable, simplifying the solution of a system of equations. It usually involves aligning coefficients and strategically adding or subtracting equations.
To eliminate a variable, multiply one or both equations by suitable numbers so that, when you add or subtract them, one of the variables cancels out.

• In this problem, the first equation is multiplied by 2. This makes the coefficient of \(x\) in both equations the same, \(8x\). The equation becomes \(8x - 6y = 22\).
• Subsequently, subtraction of the second equation from this modified first equation effectively eliminates \(x\), resulting in an equation with a single variable \(y\).

This allows you to solve for \(y\) directly, streamlining the solution process before returning to find \(x\).

Algebraic Verification

Algebraic verification involves checking that your solution satisfies all original equations. This is a critical step in solving systems of equations, ensuring that errors aren't overlooked.
Once the values of the variables are determined, substitute them back into the original equations, as a way to certify the correctness of the solution.

• Here, after determining \(x = 2\) and \(y = -1\), substitute back into the equations:
• For \(4x - 3y = 11\), check: \(4(2) - 3(-1) = 11\) which simplifies to \(8 + 3 = 11\).
• For \(8x + 4y = 12\), verify: \(8(2) + 4(-1) = 12\) which confirms \(16 - 4 = 12\).

This confirms that the solution not only matches each equation but also assures us that our calculations are correct.