Problem 29 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The solution to the system is \(\{x, y\} = \{-4, 4\}\).

Step 1: Choose a solution method

For this linear system, one can see that the first equation is already solved for \(y\). Thus, it's most efficient to use the substitution method for solving the system.

Step 2: Apply the substitution method

From the first equation, one can see that \(y = 4\). Substitute \(y = 4\) into the second equation to solve for \(x\). Thus, the second equation simplifies to \(x + 2(4) = 4\), which results in \(x = -4\).

Step 3: Verify the solution

The solution is \(\{x, y\} = \{-4, 4\}\). To verify it, substitute these values into both original equations. The first becomes \(-4 = -4\), and the second becomes \(-4 + 2(4) = 4\), verifying that both equations hold true with these solutions.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of linear equations. It is especially effective when one of the equations is already solved for one of the variables, as in the given exercise with the equation \(-y = -4\) simplifying to \(y = 4\). The basic idea of the substitution method is to substitute this simple expression of one variable into the other equation. This technique reduces the system into a single equation in one variable, making it easier to solve.
Here’s how it works:

• Identify the equation that is simplest to solve for one of its variables.
• Express this variable in terms of the other variable(s) using the chosen equation.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation to find the value of one variable.
• Substitute back to find the remaining variables.

Using the substitution method is ideal when the system is already arranged in a way that allows easy manipulation, avoiding complex calculations and reducing the risk of errors.

Linear Equations

Linear equations are fundamental components of linear systems, which are sets of equations with multiple variables, usually representing some relationships. They are termed 'linear' because they graph as straight lines on a coordinate plane. Each equation in the system represents such a line, and the solution to the system is the point(s) where these lines intersect.
Characteristics of linear equations include:

• They are in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants.
• They result in a straight line when plotted.
• Solutions can be visualized as the points of intersection of the lines represented by these equations.

In the given exercise, the linear equations are \(-y = -4\) and \(x + 2y = 4\). These equations can be solved using various techniques, such as substitution or graphing, to find where they intersect, which represents the solution of the system.

Solution Verification

Verifying the solution of a linear system is an important step to ensure the accuracy of your results. Once you obtain values for the variables, you should substitute these back into the original equations to check their correctness.
The verification process entails:

• Substitute the values of the variables back into the original equations one by one.
• Check each equation separately to ensure that they hold true, i.e., the left-hand side equals the right-hand side.
• If the values satisfy all equations in the system, the solution is correct.

For the given system, solving gives us the solution \(\{x, y\} = \{-4, 4\}\). Verification involves substituting \(x = -4\) and \(y = 4\) into both original equations. The first equation \(-y = -4\) simplifies correctly, and the second equation \(x + 2y = 4\) also holds, confirming that the solution is accurate. Regular verification like this avoids costly mistakes and solidifies understanding of the problem-solving process.