Problem 39 - 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. $$ \left\\{\begin{array}{l} 3.5 x+2.5 y=17 \\ -1.5 x-7.5 y=-33 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 2\) and \(y = 4\).

Step 1: Eliminate Decimals

To eliminate decimals, multiply the entire first equation by 10 to get rid of the decimals. Similarly, multiply the second equation by 10.\[ 35x + 25y = 170 \] \[ -15x - 75y = -330 \]

Step 2: Align Coefficients for Elimination

In this step, we align the coefficients of either \(x\) or \(y\) so that we can eliminate one variable when we add the equations. Let's eliminate \(x\) by finding a common multiple for the coefficients of \(x\).The least common multiple of 35 and 15 is 105. Multiply the first equation by 3 and the second one by 7:\[ 3(35x + 25y) = 3(170) \] \[ 7(-15x - 75y) = 7(-330) \]This gives us:\[ 105x + 75y = 510 \] \[ -105x - 525y = -2310 \]

Step 3: Add the Equations

Add the two resulting equations together so that \(x\) will be eliminated:\[ (105x + 75y) + (-105x - 525y) = 510 + (-2310) \] This simplifies to:\[-450y = -1800 \]

Step 4: Solve for Variable

Now solve for \(y\) by dividing both sides of the equation from Step 3 by \(-450\):\[ y = \frac{-1800}{-450} \] This simplifies to:\[ y = 4 \]

Step 5: Substitute to Find Second Variable

Use the value of \(y\) found in Step 4 and substitute it back into one of the original equations to solve for \(x\). Let's use the first equation:\[ 35x + 25(4) = 170 \] Simplify and solve for \(x\):\[ 35x + 100 = 170 \] \[ 35x = 70 \] \[ x = 2 \]

Step 6: Verify the Solution

To ensure the solution is correct, substitute \(x = 2\) and \(y = 4\) back into the second original equation:\[ -1.5(2) - 7.5(4) = -33 \] \[ -3 - 30 = -33 \] This holds true. Therefore, the solution is verified.

Key Concepts

Elimination Method

The elimination method is a powerful tool used in solving systems of equations. It enables you to eliminate one of the unknown variables, making it easier to solve for the other. Here's how it works:

• Begin by writing both equations in a standard form, aligning like terms vertically.
• Choose the variable to eliminate. It often depends on which variable seems easier to cancel out based on the equations found.
• To eliminate your chosen variable, make the coefficients of that variable equal in both equations. This can involve multiplying entire equations by necessary factors.
• Once coefficients are matched, add or subtract the equations. This step should eliminate one variable, leaving you with a single-variable equation to work with.

This method is particularly useful when dealing with linear equations, as it efficiently simplifies the process of finding the solution.

Decimals

Decimals can sometimes complicate solving systems of equations, especially when applying the elimination method. To streamline the process, it's often beneficial to first eliminate decimals from the equations. You can do this by multiplying every term in the equations by a power of 10 that transforms each decimal into a whole number.

• Identify the number of decimal places in the number with the most decimals in each term of the equations.
• Determine the power of 10 needed to convert these decimals into integers.
• Multiply the entire equation by this power of 10 to clear the decimals in one swoop.

These steps make the equations easier to handle mathematically, ensuring clarity and reducing potential errors during computation.

Least Common Multiple

The concept of the least common multiple (LCM) is pivotal when using the elimination method. The LCM is the smallest multiple that is common between the coefficients of a particular variable in two equations. Here's why it’s important:

• The LCM allows you to equalize coefficients across equations, facilitating cancellation of an unwanted variable when equations are added or subtracted.
• Finding the LCM involves determining the smallest number that both coefficients divide into without leaving a remainder.
• This step is crucial to ensure that multiplying equations for elimination does not unnecessarily complicate the numerics of other coefficients.

Utilizing the LCM ensures an optimal approach to eliminate variables, making the path to solution both efficient and straightforward.