Problem 26 - 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.$$ \left\\{\begin{array}{l} 6 x-5 y=7 \\ 4 x-6 y=7 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{7}{16}\) and \(y = \frac{-7}{8}\).

Step 1: Align the System of Equations

First, write the system of equations clearly, aligning the equations one below the other: \(6x - 5y = 7\) \(4x - 6y = 7\).

Step 2: Equalize Coefficients

For the addition method, we need to have the coefficients of either \(x\) or \(y\) be opposites. We'll eliminate \(x\) by making the coefficients of \(x\) in both equations equal. Multiply the first equation by 2 and the second equation by 3:\( (2)(6x - 5y) = (2)(7)\) giving \(12x - 10y = 14\) \( (3)(4x - 6y) = (3)(7)\) giving \(12x - 18y = 21\).

Step 3: Subtract Equations

Since both equations have the same \(x\) term after adjustment, subtract the second modified equation from the first modified equation. \( (12x - 10y) - (12x - 18y) = 14 - 21\)which simplifies to:\(8y = -7\).

Step 4: Solve for y

Divide both sides of the equation by 8:\(y = \frac{-7}{8}\).

Step 5: Substitute and Solve for x

Substitute \(y = \frac{-7}{8}\) back into the original first equation to solve for \(x\):\(6x - 5\left(\frac{-7}{8}\right) = 7\)which simplifies to:\(6x + \frac{35}{8} = 7\)Subtract \(\frac{35}{8}\) from both sides:\(6x = 7 - \frac{35}{8}\).Convert 7 to a fraction: \(\frac{56}{8}\), so\(6x = \frac{56}{8} - \frac{35}{8} = \frac{21}{8}\).Divide by 6:\(x = \frac{21}{8 \times 6} = \frac{7}{16}\).

Key Concepts

Addition Method

The addition method, sometimes called the elimination method, is a technique used to solve systems of linear equations. It involves combining the equations in order to eliminate one of the variables. This is done by adding or subtracting the equations to cancel out one variable, making it easier to solve for the other.

Here’s how it works:

• Choose which variable you want to eliminate, either the x or the y.
• Modify the equations so that the coefficients of this chosen variable are opposites.
• Add or subtract the equations to eliminate the chosen variable.

After removing one variable, you can solve the resulting equation for the other variable. Once you've found one variable's value, substitute it back into one of the original equations to solve for the remaining variable. This method is particularly effective when the equations are easily manipulated to create matching coefficients for elimination.

Solving Equations with Fractions

Fractions can make solving equations a bit more challenging, mainly due to the arithmetic involved. However, with a systematic approach, they can be handled efficiently. Here are some tips:

• If the equation includes fractions, consider clearing them first by multiplying both sides of the equation by a common denominator.
• This simplification helps prevent arithmetic errors and makes the equations look simpler.
• Once the fractions are cleared, proceed with solving the system using the addition or substitution method.

In our problem, we see equations becoming fraction-laden but ultimately manageable. When you solve for one variable and need to work fractions into your calculations, ensure clarity in your arithmetic to avoid mistakes. Always remember to simplify fractions when possible.

Eliminating Variables

Eliminating one variable from a system of equations simplifies the problem, narrowing down the number of equations you’re dealing with. This approach utilizes the addition method to make coefficients opposites for one of the variables, allowing for elimination.

Steps to eliminate a variable:

• Decide on which variable to eliminate from both equations.
• Adjust the coefficients by multiplying the equations with necessary constants so that they become opposites.
• Add or subtract the equations to cancel the chosen variable.

The beauty of this approach lies in its directness – once one variable is gone, you're left with a simpler single-variable equation. Solution times can be quicker as a result, reducing complex systems into more manageable pieces for solving.

Aligning Equations

Aligning equations is the first step in systematically solving a system. By writing them out clearly, one under the other, you ensure organization, which helps in accurately proceeding with more complex operations.

• Begin by writing each equation so that similar terms (like variables) are vertically aligned.
• This visual clarity makes it easier to spot opportunities for manipulation, such as multiplication, addition, or subtraction.
• Good alignment aids in identifying which coefficients to adjust, as it shows the interaction between corresponding terms clearly.

Aligning is fundamental for any method, whether it be substitution or addition, as it supports accuracy and efficiency throughout the solving process. Proper alignment acts as a simple yet powerful tool for organizing and solving equations effectively.