Problem 82 - Algebra 1: Concepts and Skills

Question

Solve the linear system. (Lessons 7.2,7.3) $$ \begin{array}{r} {2 x-y=8} \\ {2 x+2 y=2} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 3\) and \(y = -2\)

Step 1: Analyzing the equations

We have two equations: \(2x - y = 8\) and \(2x + 2y = 2\). Learn about each of the equations and identify if they can be modified for elimination.

Step 2: Preparing for elimination

In our case the equations are organized in a manner that would make elimination straightforward. But it's noticed that the second equation can be simplified by dividing all terms by 2 to get \(x + y = 1\). Now the equations are \(2x - y = 8\) and \(x + y = 1\). This makes the following steps easier.

Step 3: Applying the elimination method

Add both equations together to eliminate y, you get \(3x = 9\). This is because negative y in the first equation and positive y in the second equation will cancel each other.

Step 4: Solve for x

When you divide each side of equation \(3x=9\) by 3, you get \(x = 3\).

Step 5: Solve for y

Then, substitute \(x = 3\) back into the second original equation (or into the equation after step 2, i.e. \(x + y = 1\)) to find the value of \(y\). This gives: \(3 + y = 1\), which simplifies to \(y = 1 - 3\), so \(y = -2\)

Key Concepts

Elimination Method

The Elimination Method is a powerful tool for solving linear systems of equations. It's particularly useful when dealing with systems that can be easily manipulated to eliminate one of the variables. The goal is to add or subtract equations in the system to remove one of the variables, making it easier to solve for the remaining ones.
To use the elimination method, follow these steps:

• Firstly, align the equations so that the variables are in the same order.
• Look for opportunities to add or subtract the equations. This can help you eliminate one of the variables by combining terms.
• If necessary, you might need to multiply one or both of the equations by a constant. This helps to ensure that the coefficients of one of the variables are opposites.
• Once a variable is eliminated, solve the resulting single-variable equation.
• Finally, substitute the found value back into one of the original equations to solve for the other variable.

In this exercise, subtraction wasn't necessary after simplifying the second equation, making the application of the elimination method straightforward.

Algebraic Equations

Algebraic equations, including linear ones, are equations that represent relationships between variables and constants. They are foundational concepts in algebra, used to describe different mathematical relationships.
Understanding algebraic equations involves recognizing their structure, typically in the form of variables with coefficients plus constant terms equals some value.

• This exercise includes two algebraic equations, each with two variables, x and y.
• Each equation expresses a relationship involving these variables.
• Learning to manipulate these equations, such as simplifying or rearranging them, is key to finding solutions.

In particular, recognizing when you can simplify an equation, like dividing all terms by a common factor, can make the system easier to work with, as shown in this exercise.

Linear Equations

Linear equations are equations that graph as straight lines when plotted on a coordinate plane. They are among the simplest types of algebraic equations and are often used to model real-world situations.
Linear equations have the form:

• They usually include constant multiples of variables that are added or subtracted together, equated to a constant.
• No variables are multiplied by each other, and no variables have exponents other than one.

In this exercise, both given equations were linear because they fit this form. Solving such systems involves finding a point where both equations intersect. In our case, the process involved simplifying and comparable alignment via elimination, ultimately to find the solution \(x = 3\) and \(y = -2\), where both linear equations meet. This intersection point is the common solution to the system.