Problem 96 - Elementary Algebra

Question

$$ \left\\{\begin{array}{l} -x-12 y=-1 \\ 2 x-8 y=-6 \end{array}\right. $$$$ \left\\{\begin{array}{l} -x-12 y=-1 \\ 2 x-8 y=-6 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is

Step 1: Simplify the system

First, write down the given system of equations: - (i) - (ii)

Step 2: Multiply equation (ii)

To eliminate x, multiply equation (ii) by 1: ii * 1:

Step 3: Add the equations

Add (i) and (ii) to eliminate x:

Step 4: Solve for y

Simplify the equation obtained from step 3 to find the value of y:

Step 5: Substitute y in equation (i)

Substitute the value of y into equation (i) to find x:

Step 6: Verify the solution

Substitute the values of and y into both original equations to check their correctness: and

Key Concepts

Elimination Method

The Elimination Method is a popular technique for solving a system of linear equations. The goal is to eliminate one of the variables by adding or subtracting the equations, thereby reducing the system to a single equation with one variable. Let's understand how it works:

• First, align the equations in a way that you can easily add or subtract them.
• Multiply one or both equations to get coefficients of one variable to be the same (or opposites).
• Add or subtract the equations to eliminate one variable.
• Solve the resulting single-variable equation.
• Substitute the solution back into one of the original equations to find the other variable.

For example, if you have the system \(\begin{array}{l} -x - 12y = -1 \ 2x - 8y = -6 \ \end{array}\), you can eliminate one variable by manipulating and adding/subtracting these equations.

Substitution Method

The Substitution Method is another efficient technique for solving systems of linear equations. This method involves isolating one variable in one of the equations and substituting this expression into the other equation. Steps to use the substitution method include:

• Solve one of the equations for one variable in terms of the other.
• Substitute this expression into the other equation.
• Solve the resulting equation for the second variable.
• Use the value found to solve for the first variable in the isolated equation.

For example, if we have the system \(\begin{array}{l} -x - 12y = -1 \ 2x - 8y = -6 \ \end{array}\), you can solve one of the equations for x or y and substitute this into the other equation.

Solving Equations

Solving equations is at the heart of finding unknown values in a system of linear equations. The primary goal is to find the values of variables that satisfy all equations in the system. Let's break it down:

• Identify the variables and equations involved.
• Use either the Elimination or Substitution method to isolate one variable.
• Simplify the resulting single-variable equation.
• Find the value of this variable.
• Substitute back into one of the original equations to find the other variable.
• Verify your solutions by substituting the values back into both original equations.

For the given problem, solving equations step-by-step helps in ensuring each variable's value is correctly identified. Remember to always check your solutions for consistency.