Problem 9 - Intermediate Algebra

Question

For the following system, clear the equations of any fractions or decimals and write each equation in \(A x+B y+C z=D\) form. $$ \left\\{\begin{array}{l} x+y=3-4 z \\ 0.7 x-0.2 y+0.8 z=1.5 \\ \frac{x}{2}+\frac{y}{3}-\frac{z}{6}=\frac{2}{3} \end{array}\right\\{ $$

Step-by-Step Solution

Answer

The system in standard form is: 1. \( x + y + 4z = 3 \) 2. \( 7x - 2y + 8z = 15 \) 3. \( 3x + 2y - z = 4 \).

Step 1: Clear the Fractions in Equation 1

The first equation is already without fractions or decimals. Rewrite it for clarity: \[ x + y + 4z = 3 \] which is already in the form \( Ax + By + Cz = D \).

Step 2: Clear the Decimals in Equation 2

To eliminate decimals in the second equation, multiply every term by 10: \[ 0.7x - 0.2y + 0.8z = 1.5 \] becomes \[ 7x - 2y + 8z = 15 \].

Step 3: Clear the Fractions in Equation 3

To clear fractions, find a common denominator. Here, the common denominator for 2, 3, and 6 is 6. Multiply every term by 6: \[ \frac{x}{2} + \frac{y}{3} - \frac{z}{6} = \frac{2}{3} \] becomes \[ 3x + 2y - z = 4 \].

Step 4: Write the System in \(Ax + By + Cz = D\) Form

Now, compile the simplified equations into the standard form:1. \( x + y + 4z = 3 \)2. \( 7x - 2y + 8z = 15 \)3. \( 3x + 2y - z = 4 \).

Key Concepts

Fraction Elimination

Fraction elimination is an essential skill when solving systems of equations. Fractions can make equations cumbersome and challenging to solve. By eliminating fractions, you simplify the equation, making it easier to handle.

To remove fractions from an equation, follow these steps:

• Identify the denominators in the equation. This will help you determine the least common denominator (LCD) for all terms containing fractions.
• Multiply every term in the equation by the LCD. This action eliminates the denominators by turning fraction terms into whole numbers.
• Simplify the resulting equation. Ensure every term is reduced to its simplest form.

For example, in the equation \( \frac{x}{2} + \frac{y}{3} - \frac{z}{6} = \frac{2}{3} \), the LCD is 6. Multiplying every term by 6 transforms it into \( 3x + 2y - z = 4 \). Eliminating fractions clarifies the path toward solving the system efficiently.

Decimal Elimination

Decimals can also obstruct the clarity and simplicity of equations. Just like with fractions, eliminating decimals achieves a cleaner equation form which is easier to manipulate.

Here's how to eliminate decimals from an equation:

• Determine the number of decimal places in the terms involved. All decimals should have the same number of decimal places for uniformity.
• Multiply every term in the equation by a power of 10 necessary to remove the decimals. For example, using 10 for one decimal place, 100 for two decimal places, and so on.
• Reform the resulting equation, checking for any simplifications needed.

Consider the equation \( 0.7x - 0.2y + 0.8z = 1.5 \). Here, you’d multiply by 10 to remove the decimals, transforming it into \( 7x - 2y + 8z = 15 \). This maneuver solidifies the equation into a more straightforward format.

Systems of Equations

Systems of equations occur frequently in algebra, representing scenarios with multiple relationships between variables. These systems often look to find values for each variable that satisfy all equations simultaneously.

A system is typically composed of two or more equations, each containing common variables. When solving these systems, it's crucial to present them in a standard form. The form \( Ax + By + Cz = D \) is standard because it aligns coefficients and variables neatly, facilitating easier comparison and solution.

To bring a system into this form, approach each equation to remove fractions and decimals, and rearrange terms as needed. For instance, a given system like this:
- \( x + y + 4z = 3 \)
- \( 7x - 2y + 8z = 15 \)
- \( 3x + 2y - z = 4 \)
It provides a clear pathway for solving the system using substitution or elimination methods, ensuring consistency in solving multi-variable problems.