Problem 34 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals. $$ \left\\{\begin{array}{l} -9(x+3)=8 y \\ 3 x-3 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{17}{3}\) and \(y = 3\).

Step 1: Expand the First Equation

Start by expanding the first equation to remove the parentheses. The equation is \(-9(x+3)=8y\). Distribute the \(-9\) across the terms inside the parentheses to get \(-9x - 27 = 8y\).

Step 2: Align Both Equations

Write both equations one under the other:1. \(-9x - 27 = 8y\)2. \(3x - 3y = 8\)

Step 3: Eliminate Fractions/Decimals

Since there are no fractions or decimals in the given equations, proceed to the next step of elimination using addition.

Step 4: Make Coefficients Opposite

To eliminate \(x\), multiply the second equation by 3 to align the coefficients of \(x\): \(3 \times (3x - 3y = 8) \Rightarrow 9x - 9y = 24\).

Step 5: Add the Equations

Add the modified equations: \(-9x - 27 = 8y\) and \(9x - 9y = 24\).This results in:\(0x - 27 = 8y - 9y + 24\), which simplifies to \(-27 = -y + 24\).

Step 6: Solve for y

Rearrange \(-27 = -y + 24\) to solve for \(y\): Add \(y\) and \(27\) to both sides to get:\(-y = -27 + 24\).Simplify to get:\(-y = -3\), which simplifies to \(y = 3\).

Step 7: Substitute y Back into an Equation

Substitute \(y = 3\) into the second original equation \(3x - 3y = 8\).This becomes \(3x - 3(3) = 8\).Simplify to: \(3x - 9 = 8\).

Step 8: Solve for x

Add 9 to both sides: \(3x = 17\).Divide by 3: \(x = \frac{17}{3}\).

Step 9: State the Solution

The solution to the system of equations is \(x = \frac{17}{3}\) and \(y = 3\).

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. These equations are like teammates working together to lock down the values for the variables. Thus, each equation in the system gives us a piece of the puzzle. By working with all of the equations together, we can solve for all the variables involved.

• For a system of equations with two variables, the equations usually represent lines on a graph.
• The solution to the system is where these lines intersect, meaning they share the same coordinates.
• There are several ways to find the solution, including graphing, substitution, and addition (elimination) method.

Understanding systems of equations is crucial in algebra since it enables us to model real-world situations and find solutions that are meaningful.

Solving Equations

The goal of solving equations is to find the value or values of the unknowns that satisfy each equation in the system. Solving an equation involves:

• Identifying the unknown variables.
• Performing operations to isolate these variables on one side of the equation.

In our example problem,

• We are solving for two variables: \(x\) and \(y\).
• By isolating one variable at a time, we can gradually uncover their values.

Each step of solving the system brings us closer to understanding how the two equations relate to one another and where exactly they intersect.

Algebraic Manipulation

Algebraic manipulation is a fancy term for the strategies we use when dealing with algebraic expressions and equations. It involves rearranging and combining terms through addition, subtraction, multiplication, and division to reach a solution. In this context, algebraic manipulation helps us to align coefficients of variables or clear fractions and decimals. Here are some common techniques:

• Expanding expressions, as with the equation \(-9(x+3) = 8y\) becoming \(-9x - 27 = 8y\).
• Adjusting equations by multiplying or dividing all terms. For example, multiplying the second equation by 3 to create \(9x - 9y = 24\).
• Rearranging the terms in an equation to isolate and solve for a variable.

Algebraic manipulation is essential for clearly expressing mathematical relationships and solving equations logically.

Elimination Method

The elimination method, also known as the addition method, is a powerful technique used to solve systems of equations. Its strength lies in its ability to "eliminate" one variable by aligning coefficients, allowing us to solve for the other variable. Here's how it works:

• Firstly, you adjust the coefficients of one variable so they are opposites (like in \(-9x\) and \(9x\) in the example).
• Next, you add the equations together to cancel out that variable, simplifying the system to one equation with one unknown.
• Then, solve the simplified equation for the remaining variable.

In our given example, aligning coefficients led to eliminating \(x\), allowing us to solve for \(y\), followed by back-substitution to find \(x\). The elimination method is especially useful when dealing with linear equations, as it tends to be more straightforward and less prone to errors compared to other methods like substitution.