Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{aligned}&\frac{x}{5}+3 y=31\\\&2 x-\frac{y}{5}=8\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 5\), \(y = 10\).

Step 1: Eliminate Fractions

To eliminate fractions from the equations, find a common multiple for each equation's denominators. In this case, multiply the first equation by 5 and the second equation by 5.For the first equation:\[ 5 \left( \frac{x}{5} \right) + 5(3y) = 5 \times 31 \]This simplifies to:\[ x + 15y = 155 \]For the second equation:\[ 5(2x) - 5 \left( \frac{y}{5} \right) = 5 \times 8 \]This simplifies to:\[ 10x - y = 40 \]

Step 2: Line Up Equations and Decide Variable to Eliminate

We now have the equations:\[ x + 15y = 155 \]\[ 10x - y = 40 \]To eliminate a variable, we'll choose to eliminate \(y\). Multiply the second equation by 15 to match the coefficient of \(y\) in the first equation.

Step 3: Adjust Second Equation for Elimination

Multiply the second equation by 15:\[ 15(10x - y) = 15(40) \]This simplifies to:\[ 150x - 15y = 600 \]

Step 4: Add Equations to Eliminate Variable

Now add the adjusted second equation to the first equation:\[ (x + 15y) + (150x - 15y) = 155 + 600 \]The terms \(15y\) and \(-15y\) cancel out, giving:\[ 151x = 755 \]

Step 5: Solve for x

Divide both sides by 151 to solve for \(x\):\[ x = \frac{755}{151} \]This simplifies to:\[ x = 5 \]

Step 6: Substitute x Back to Find y

Substitute \(x = 5\) back into one of the original equations to find \(y\). Use the equation \(x + 15y = 155\):\[ 5 + 15y = 155 \]Subtract 5 from both sides:\[ 15y = 150 \]Divide by 15:\[ y = 10 \]

Step 7: Verify Solution

Check the solution \(x = 5\) and \(y = 10\) in both original equations.For \(\frac{x}{5} + 3y = 31\):\[ \frac{5}{5} + 3(10) = 1 + 30 = 31 \] (True)For \(2x - \frac{y}{5} = 8\):\[ 2(5) - \frac{10}{5} = 10 - 2 = 8 \] (True)Since both equations are satisfied, the solution is verified.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. Its main goal is to eliminate one of the variables, allowing for simpler equations to solve.

• First, we manipulate the equations to align coefficients of one variable for both equations.
• Then, by adding or subtracting these equations, we eliminate one of the variables.
• Once one variable is eliminated, we solve the resulting equation for the remaining variable.

This method hinges on the principle that equations can be combined without changing their solutions. In our original problem, this involved aligning the coefficients of the variable \(y\) by multiplying one equation, then adding to eliminate \(y\) entirely.

Fractions in Equations

Fractions often appear in equations and can complicate solving processes. To handle them efficiently, first eliminate fractions to simplify equations.

• Identify a common multiple of all denominators in the equation.
• Multiply those equations by this multiple to clear out fractions.
• Ensure that each term of the equation is scaled appropriately.

In our exercise, we multiplied each side of the equations by 5 to eliminate denominators. This reformulated the problem into simpler equations without fractions, easing the process of applying the elimination method.

Solving Linear Equations

Solving linear equations typically involves isolating variables through arithmetic operations. Once simplified, calculate the value of the variable.

• Use addition or subtraction to get rid of constants near the variable.
• Employ multiplication or division to isolate the variable completely.
• Substitute back to check consistency and solve for the second variable in systems.

In our problem, after eliminating \(y\), we had a linear equation in one variable, \(x\), which we solved by basic division, given \(x = 5\). We then substituted back to find \(y\).

Verification of Solutions

Verification is key in ensuring the solution is correct. After calculating the variables, re-substitute them into original equations to confirm correctness.

• Check each original equation with calculated variable values.
• Ensure both sides of the equations equate correctly.
• If verified, the solution is correct and complete.

In our step-by-step solution, we verified that \(x = 5\) and \(y = 10\) satisfied both given equations, confirming the correctness and accuracy of our work. Verification reassures us that the operations performed were correct.