Problem 21 - Algebra 1

Question

Choose a method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &x-2 y=4\\\ &6 x+2 y=10 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = 2 \) and \( y = -1 \).

Step 1: Choosing the Method to Solve the System

Analyze the two equations given. The best approach to solve this system is the method of addition or subtraction. Considering the coefficient of \( y \) in both equations, it is evident that adding the two equations will eliminate \( y \).

Step 2: Add the Two Equations

Add both equations together. \( (x-2y) + (6x+2y) = 4 + 10 \) which simplifies to \( 7x = 14 \).

Step 3: Solve for x

Solve for \( x \) by dividing the equation by 7. \( x = 14 / 7 \) which simplifies to \( x = 2 \).

Step 4: Substitute \( x = 2 \) into the Equation

Substitute \( x = 2 \) into the first equation. \( 2 - 2y = 4 \) which simplifies to \( -2y = 4 - 2 \). This simplifies further to \( -2y = 2 \).

Step 5: Solve for y

Solve the equation for \( y \) by dividing the equation by \( -2 \). \( y = 2 / -2 \) which simplifies to \( y = -1 \).

Key Concepts

Addition Method

One of the most efficient strategies to tackle systems of linear equations is the addition method, also known as the elimination method. This technique involves manipulating the equations to eliminate one of the variables, making it simpler to solve. In the provided exercise, this method is chosen because the coefficients of one of the variables (\( y \) in this case) are particularly convenient for elimination.
By adding or subtracting the equations, we aim to cancel out one of the variables. In practice, here’s how the addition method works:

• Align both equations so that corresponding variables and constants line up vertically.
• Add or subtract the equations directly. This operation should result in one variable being eliminated from one of the equations.
• The result is a new equation in which only one variable remains, allowing for straightforward solving.

In our example, after adding the two equations, the variable \( y \) is eliminated, simplifying the system to a single equation \( 7x = 14 \). This makes solving for \( x \) straightforward.

Substitution Method

Another valuable method for solving linear systems is the substitution method. This approach is particularly useful when one equation in the system enables easy isolation and expression of one variable in terms of the other.
The substitution method can be succinctly outlined with the following steps:

• Solve one of the equations for one of the variables (for example, express \( x \) in terms of \( y \)).
• Substitute this expression into the other equation.
• This substitution yields a new equation with one variable, which is easier to solve.
• After solving the single-variable equation, back-substitute the value into one of the original equations to find the second variable.

Though not used directly in the provided example, substitution is a powerful technique – especially when equations are not easily manipulated by addition or when no matching coefficients are present.

Linear Equations

Linear equations form the basis of linear systems, which are sets of two or more linear equations. These equations express relationships involving constants and linear terms (terms like \( x \) and \( y \)) but not exponents or complex powers.
A standard linear equation takes the form:

• \( ax + by = c \)

Where \( a \), \( b \), and \( c \) are constants. The task when solving a system is to find values of \( x \) and \( y \) that make all equations true simultaneously.
In our example, we work with two linear equations:

• \( x - 2y = 4 \)
• \( 6x + 2y = 10 \)

These equations can be visualized as lines on a plane, and solving them means finding the intersection of these lines, which represents the solution \((x, y)\). This solution tells us the values that satisfy both conditions posed by the linear equations simultaneously. Linear equations not only help in academic problems but also model real-world scenarios such as predicting financial trends or calculating forces in physics.