Problem 1 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. $$ \left\\{\begin{array}{l} 3 x+y=5 \\ 6 x-y=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 1\) and \(y = 2\).

Step 1: Add the Equations

To eliminate one of the variables, add the two equations together. Start with the given system:\[ 3x + y = 5 \ 6x - y = 4 \] Adding these two equations will cancel out the 'y' terms because \(y - y = 0\). Therefore, we get: \[ 9x = 9 \]

Step 2: Solve for x

Now solve the equation \(9x = 9\) for \(x\) by dividing both sides by 9: \[ x = \frac{9}{9} = 1 \] Thus, \(x\) is equal to 1.

Step 3: Substitute x into the First Equation

Substitute \(x = 1\) back into the first equation to solve for \(y\). The first equation was: \[ 3x + y = 5 \] Substitute \(x = 1\): \[ 3(1) + y = 5 \] Simplify: \[ 3 + y = 5 \]

Step 4: Solve for y

To find \(y\), subtract 3 from both sides: \[ y = 5 - 3 \] \[ y = 2 \] Thus, \(y\) is equal to 2.

Key Concepts

Addition Method

The addition method is a popular technique for solving systems of equations. It is sometimes known as the "method of elimination" because it focuses on eliminating one variable by adding the equations together. The key idea is to line up the equations so that when they are added, one of the variables cancels out. This allows you to discover the value of the second variable.
Here's how it works: you align two equations vertically and ensure that one of the variables has equal but opposite coefficients. By adding them, you can "eliminate" this variable, making it much easier to solve for the remaining variable. This method is especially useful when both equations are already set up nicely, as in our system with the equations:

• \(3x + y = 5\)
• \(6x - y = 4\)

By simply adding these two equations, we effectively eliminated the variable \(y\), leaving us with a single variable \(x\), which makes solving simpler.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables which are solved together. This means that the solution is a pair (or more) of values that satisfy all equations at the same time. In this context, we are looking for values of \(x\) and \(y\) that solve both equations in the system:

• \(3x + y = 5\)
• \(6x - y = 4\)

The goal is to find a common solution for both equations, which represents the intersection point of the two lines when graphed. For our problem, this is where the addition method comes into play, simplifying the solution of these equations at the same time. Using the approach, we quickly discover that the solution is \(x = 1\) and \(y = 2\).

Elimination Method

The elimination method is essentially another name for the addition method, as the process involves eliminating one of the variables to simplify the equations. In our case, we want to eliminate the variable \(y\) because the equations are:

• \(3x + y = 5\)
• \(6x - y = 4\)

Here, by looking at the coefficients of \(y\) in these equations (+1 in one equation and -1 in the other), we notice they will cancel each other out when added. This leaves: \(9x = 9\), making it easy to solve for \(x\). We then substitute this back into one of the original equations to find \(y\). This method is great for systems where the coefficients can be manipulated or are already aligned to cancel out directly.

Algebraic Solution

Once you have eliminated one variable using the elimination method, you're left with a single-variable equation. This significantly reduces complexity and allows for straightforward algebraic manipulation to solve for the unknown. In our system, with the equation reduced to \(9x = 9\), we can quickly find that \(x = 1\) simply by dividing both sides of the equation by 9.
With the value of \(x\) known, the next logical step is to substitute this value back into one of the original equations to find \(y\). Using \(3x + y = 5\), we substitute \(x = 1\), leading to:

• \(3(1) + y = 5\)

which simplifies to \(y = 2\). By handling these steps, students can realize that the solution to the equations is \((x, y) = (1, 2)\), representing the point where the lines would intersect graphically.