Problem 43 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{l} \frac{w}{2}-\frac{t}{5}=11 \\ \frac{w}{8}-\frac{t}{9}=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \(w = 40\) and \(t = 45\).

Step 1 - Clear Fractions in the First Equation

Start by clearing the fractions in the first equation. Multiply both sides of the equation by the least common multiple (LCM) of the denominators 2 and 5, which is 10:\(10 \times \frac{w}{2} - 10 \times \frac{t}{5} = 10 \times 11\)\(5w - 2t = 110\)

Step 2 - Clear Fractions in the Second Equation

Now clear the fractions in the second equation by multiplying both sides by the LCM of the denominators 8 and 9, which is 72:\(72 \times \frac{w}{8} - 72 \times \frac{t}{9} = 72 \times 0\)\(9w - 8t = 0\)

Step 3 - Choose a Method to Solve the System

We will use the elimination method. First, align the equations we obtained:1) \(5w - 2t = 110\)2) \(9w - 8t = 0\)

Step 4 - Eliminate One Variable

Multiply the first equation by 4 to align the coefficients of \(t\):\(4(5w - 2t) = 4(110)\)\(20w - 8t = 440\)Now subtract the second equation from this new equation:\((20w - 8t) - (9w - 8t) = 440 - 0\)\(20w - 9w = 440\)\(11w = 440\)

Step 5 - Solve for \(w\)

Divide both sides of the equation by 11:\(w = \frac{440}{11}\)\(w = 40\)

Step 6 - Substitute \(w\) to Find \(t\)

Substitute \(w = 40\) into the second equation to solve for \(t\):\(9(40) - 8t = 0\)\(360 - 8t = 0\)Rearrange the equation:\(360 = 8t\)Divide both sides by 8:\(t = \frac{360}{8}\)\(t = 45\)

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations. It involves combining the equations in such a way that one of the variables gets canceled out, making it easier to solve for the remaining variable.

Here's how to use the elimination method step-by-step:

• Align the equations one above the other.
• Choose a variable to eliminate. This could be either variable, but often it's easiest to choose the one that requires the least manipulation.
• Adjust the equations by multiplying them by appropriate values so that the coefficients of the chosen variable become equal in both equations but with opposite signs.
• Subtract or add the equations to eliminate the chosen variable.
• Solve the resulting single-variable equation.
• Substitute the found value back into one of the original equations to find the second variable.

This method is particularly effective when the coefficients of one of the variables are easily alignable through multiplication and subtraction.

Fractions in Equations

Fractions can make solving equations cumbersome, so it often helps to clear them out early in the problem. You can do this by finding the least common multiple (LCM) of the denominators and multiplying through by this number.

Step-by-step process to clear fractions:

• Identify all the denominators in the equation.
• Calculate the LCM of these denominators.
• Multiply every term in the equation by the LCM. This will clear the fractions.
• Simplify the resulting equation to standard linear form.

For example, in our given problem: \(\frac{w}{2} - \frac{t}{5} = 11\) can be cleared of fractions by multiplying every term by 10, the LCM of 2 and 5. This gives us \(5w - 2t = 110\). This step simplifies the equation, making it easier to use the elimination method.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of them.

Finding the LCM is important when you need to clear fractions from equations. Here's a quick way to find the LCM:

• List the prime factors of each number.
• Identify the highest power of each prime factor that appears in these lists.
• Multiply these highest powers together to find the LCM.

For example, to clear the fractions in the equation \(\frac{w}{2} - \frac{t}{5} = 11\), you need the LCM of 2 and 5. Since 2 and 5 are both prime and have no common factors, the LCM is simply their product: \(2 \times 5= 10\).

By multiplying each term in the equation by 10, you remove the fractions, making the equation more manageable. This technique proves useful in simplifying equations for further steps, such as applying the elimination method.