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Chapter 7: Problem 47

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 3 x=3+2 y \\ -\frac{4}{3} x+y=\frac{1}{3} \end{array} $$

### Short Answer

Expert verified

The solution is \(x = 1\) and \(y = 0\). The solution checks out.

## Step by step solution

01

## - Rewrite the Equations in Standard Form

Rewrite both equations in the form Ax + By = C.The first equation: \(3x = 3 + 2y\) becomes \(3x - 2y = 3\)The second equation: \(-\frac{4}{3}x + y = \frac{1}{3}\) remains unchanged.

02

## - Eliminate Fractions

To eliminate fractions in the second equation, multiply every term by 3:\(-\frac{4}{3}x + y = \frac{1}{3}\) becomes \(-4x + 3y = 1\).

03

## - Combine Both Equations

Now the system is:\(3x - 2y = 3$$-4x + 3y = 1\)We will use the elimination method to solve this system.

04

## - Make the Coefficients of y the Same

To eliminate one variable, we need to make the coefficients of y the same. Multiply the first equation by 3 and the second equation by 2:\(3(3x - 2y) = 3(3)\) becomes \(9x - 6y = 9$$2(-4x + 3y) = 2(1)\) becomes \(-8x + 6y = 2\)

05

## - Add the Equations

Add the modified equations to eliminate y:\((9x - 6y) + (-8x + 6y) = 9 + 2\)This simplifies to:\(x = 11\)

06

## - Solve for y

Substitute \(x = 1\) back into one of the original equations to solve for y. Using \(3x - 2y = 3\):\(3(1) - 2y = 3\)Simplify and solve for y:\(3 - 2y = 3$$-2y = 0$$y = 0\)

07

## - Check the Solution

Substitute \(x = 1\) and \(y = 0\) back into the original equations to check the solution.First equation: \(3x = 3 + 2y$$3(1) = 3 + 2(0)\) is true.Second equation: \(-\frac{4}{3}x + y = \frac{1}{3}$$-\frac{4}{3}(1) + 0 = \frac{1}{3}\) is true.

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Solving Systems of Equations

When solving systems of equations, you deal with two or more equations that share the same variables. Typically, the goal is to find the values of these variables that satisfy all the given equations at once.

There are different methods to solve such systems, and some popular ones include:

- Graphical Method
- Substitution Method
- Elimination Method

In this case, we'll concentrate on the elimination method.

The main idea behind the elimination method is to manipulate the equations in such a way that adding (or subtracting) them eliminates one of the variables.

This simplifies the system to a single equation with one unknown, which is easier to solve.

###### Elimination Method

The elimination method is an effective way to solve systems of equations, especially when the equations are already in standard form. Here's a step-by-step approach:

1. **Rewrite in Standard Form**: Ensure each equation is in the form Ax + By = C. This means all variable terms are on one side, and constants are on the other side.

For example:

- Original: \(3x = 3 + 2y\)
- Standard Form: \(3x - 2y = 3\)

2. **Eliminate Fractions:** If any equations have fractions, it's helpful to clear them by multiplying each term by the least common denominator.

For example, the second equation: \(-\frac{4}{3}x + y = \frac{1}{3}\) becomes \(-4x + 3y = 1\) after multiplying by 3.

3. **Match Coefficients:** Adjust the coefficients of one of the variables so they can cancel each other when the equations are added or subtracted.

For instance, we can multiply the first equation by 3 to get \(9x - 6y = 9\) and the second by 2 to get \(-8x + 6y = 2\).

4. **Add or Subtract Equations:** Finally, add or subtract the modified equations to eliminate one variable, leaving a simpler equation to solve. In our example, adding gives us \(x = 11\).

This new equation can then be used to find the remaining variable.

###### Algebraic Equations

Algebraic equations are mathematical statements that involve variables and constants connected by equality signs. They form the basis of the systems we are solving.

Here's a quick rundown:

- **Univariate Equations**: Involving one variable (e.g., \(3x + 2 = 11\))
- **Multivariate Equations**: Involving two or more variables (e.g., \(2x + y = 5\))

When dealing with systems of algebraic equations, it’s crucial to ensure

1. **Consistency**: The system has at least one solution.

2. **Compatibility**: The relationships described by the equations must coexist.

3. **Simplification**: Many times equations need simplification (like eliminating fractions) to make them easier to manipulate.

Working through each step methodically helps in solving the system accurately. For example, once you find \(x = 11\), you substitute this back into one of the original equations to find the value of \(y\). This process ensures that the values derived satisfy all original equations.

After solving, always double-check your solutions by plugging them back into the original equations to verify their correctness.

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