Problem 42 - 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. $$ \left\\{\begin{array}{l} 0.04 x-0.05 y=0.105 \\ 0.2 x-0.6 y=1.05 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \(x = 0.75\) and \(y = -1.5\).

Step 1: Clear Decimals

To eliminate decimals, multiply the first equation by 100 and the second equation by 10:1. \( 0.04x - 0.05y = 0.105 \rightarrow 4x - 5y = 10.5 \)2. \( 0.2x - 0.6y = 1.05 \rightarrow 2x - 6y = 10.5 \)

Step 2: Prepare for Elimination

To eliminate one of the variables, let's target eliminating \(x\). We do this by aligning the coefficients of \(x\) from the two equations. Leave the first equation unchanged and multiply the second equation by 2 to match coefficients:1. \( 4x - 5y = 10.5 \)2. \( 4x - 12y = 21 \)

Step 3: Subtract the Equations

Subtract the second equation from the first to eliminate \(x\):\[ (4x - 5y) - (4x - 12y) = 10.5 - 21 \]Simplify to: \( 7y = -10.5 \)

Step 4: Solve for y

Isolate \(y\):\[ y = \frac{-10.5}{7} \]\( y = -1.5 \)

Step 5: Solve for x

Substitute \(y = -1.5\) back into the original simplified first equation:\( 4x - 5(-1.5) = 10.5 \)Simplify and solve for \(x\):\( 4x + 7.5 = 10.5 \)\( 4x = 3 \)\( x = 0.75 \)

Step 6: Verify the Solution

Substitute \(x = 0.75\) and \(y = -1.5\) back into the original equations to verify:1. \( 0.04(0.75) - 0.05(-1.5) = 0.105 \rightarrow 0.03 + 0.075 = 0.105 \) which is true.2. \( 0.2(0.75) - 0.6(-1.5) = 1.05 \rightarrow 0.15 + 0.9 = 1.05 \) which is true. The solution is verified.

Key Concepts

Addition Method

The addition method is a powerful technique for solving systems of linear equations where we aim to eliminate one of the variables by aligning coefficients. This method requires adding or subtracting entire equations to make the coefficient of one variable zero.

Here's how it works:

• First, we manipulate the equations so that adding or subtracting them cancels out one variable. This is done by multiplying the equations by appropriate numbers.
• Once one variable is eliminated, we solve for the remaining variable.
• Finally, we substitute the value back into one of the original equations to find the second variable.

The addition method is straightforward when equations are presented with integers, but it might require initial modifications, such as clearing decimals or fractions, to proceed smoothly.

Solving Systems of Equations

In mathematics, solving systems of equations involves finding values for the variables that satisfy all equations simultaneously.

Here are common methods to solve them:

• Graphical Method: Intersecting lines represent the solution, though this method can be less accurate.
• Substitution Method: Solve one equation for a variable, then substitute this value into the other equation. This method might become cumbersome with complex equations.
• Addition (Elimination) Method: Align and eliminate a variable by addition or subtraction of equations. It's especially useful for linear equations.

The choice of method can depend on the given equations and the presence of difficult coefficients like decimals or large numbers. With practice, recognizing which method will be fastest and easiest to apply comes naturally.

Decimal Elimination

Decimals can make solving equations more cumbersome due to their complication in calculations. Therefore, using decimal elimination simplifies the equations.

This process involves multiplying each term of the equation by a power of ten to convert decimal coefficients into whole numbers.

• Identify the highest place value of the decimals in the equations.
• Multiply every term in the equation by a power of ten that corresponds to this place value, effectively eliminating the decimal and easing calculations.
• Proceed with further simplifications or use other methods like the addition method.

For example, decimals like 0.04x or 0.2x would be multiplied by 100 and 10, respectively, to convert them into 4x and 2x, simplifying the procedure and reducing computational errors.