Problem 2 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. $$ \left\\{\begin{array}{l} 4 x+y=13 \\ 2 x-y=5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (3, 1)\).

Step 1: Write the system of equations

The system of equations given is:\[\begin{align*}4x + y &= 13 \2x - y &= 5\end{align*}\]This is already structured to apply the addition method.

Step 2: Add the two equations

To eliminate one of the variables, we add the two equations together. Notice that when adding the two equations, the terms with \(y\) will cancel each other out:\[\begin{align*}(4x + y) + (2x - y) &= 13 + 5\4x + 2x + y - y &= 18 \6x &= 18\end{align*}\]

Step 3: Solve for \(x\)

Now solve the resulting single-variable equation for \(x\):\[6x = 18\]Divide both sides by 6:\[x = \frac{18}{6} = 3\]So, \(x = 3\).

Step 4: Substitute \(x\) back into one of the original equations

To find \(y\), substitute \(x = 3\) back into one of the original equations. Let's use the first equation:\[4(3) + y = 13\]Simplify and solve for \(y\):\[12 + y = 13 \y = 13 - 12 \y = 1\]

Step 5: State the solution

The solution to the system of equations is \(x = 3\) and \(y = 1\). Therefore, the ordered pair \((3, 1)\) satisfies both equations.

Key Concepts

Solving Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. The main goal in solving a system is to find values for each variable that make all the equations true simultaneously. In the exercise, the system of equations is:

• \(4x + y = 13\)
• \(2x - y = 5\)

These are called linear equations, as the highest power of their variables is one. There are various methods to solve systems of equations, including substitution, graphing, and elimination (also known as the addition method). Each of these methods aims to isolate the variables to determine their values.The addition method, used in the solution, is particularly effective when the goal is to eliminate one variable quickly and directly. The two main objectives when solving these systems are: - Isolating each variable independently.- Ensuring that the solution satisfies all the given equations.

Eliminating Variables

Eliminating variables is a key strategy in solving systems of equations, particularly when using the addition method. In our exercise, the goal was to get rid of one of the variables to simplify the equations into a single variable equation.The original equations are:

• \(4x + y = 13\)
• \(2x - y = 5\)

By adding these two equations, the \(y\) terms cancel out:

• \((4x + y) + (2x - y) = 13 + 5\)
• After simplification, this becomes \(6x = 18\)

This cancellation process simplifies the system by reducing it to a single equation in one variable, making it much easier to solve. The purpose of eliminating variables is to isolate one variable, making it straightforward to find its value. This method is particularly useful when equations are set up to naturally allow for a specific variable cancellation when summed or subtracted.

Substitution Method

The substitution method is another popular technique for solving systems of equations, differing from the addition method mainly in its approach to isolating variables. It involves solving one equation for one variable and then substituting that expression into another equation.After solving our simplified equation from the addition method, we found that \(x = 3\). We then used the substitution method, albeit indirectly from the addition method solution. To find \(y\), we substituted \(x = 3\) back into one of the initial equations:

• \(4x + y = 13\)
• Substitute \(x = 3\) to get \(4(3) + y = 13\)
• Solving this gives \(y = 1\)

In this method, the focus is on expressing the variable through one of the equations first, making it very effective when equations present a straightforward substitution opportunity from rearranging.

Linear Equations

Linear equations are the foundation of the exercise and understanding these is essential to solving systems effectively. A linear equation in two variables, like \(4x + y = 13\), describes a straight line when graphed.Several characteristics define linear equations:

• They can be written in the standard form \(Ax + By = C\)
• Both variables are raised only to the power of one
• They have no products or functions involving the variables (such as \(xy\) or \(x^2\))

Understanding these properties helps in visualizing or conceptualizing how they interact, especially in systems. When solving, knowing that a linear equation is represented as a line helps in foreseeing solutions that equal intersection points between the lines of multiple equations. Thus, grasping the linear nature is crucial for solving and interpreting the systems like in our original problem.