Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{aligned} \frac{2 x+5}{2}+\frac{y-1}{3} &=-1 \\ 2 x-y &=12 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -18.5, y = -49\)

Step 1: Simplify Equations

Rewrite the first equation to make it easier to work with. \[2x+5 +\frac{2}{3}(y-1) =-1\]When simplified, we obtain\[2x + 5 + \frac{2}{3}y - \frac{2}{3} = -1\]This further simplifies to\[2x + \frac{2}{3}y + \frac{13}{3} = 0\]Multiply each term by 3 to get rid of the fractions:\[6x + 2y + 13 = 0\]Therefore the system of equation becomes:\[\left\{\begin{aligned}6x+ 2y + 13 & = 0\2x - y & = 12\end{aligned}\right.\]Multiply the second equation by 2 so that after subtraction, we can eliminate y:\[\left\{\begin{aligned}6x+ 2y + 13 & = 0\4x - 2y & = 24\end{aligned}\right.

Step 2: Eliminate Variable

Now we will subtract the second equation from the first one:\[(6x+2y+13) - (4x-2y) = 0 - 24\]It simplifies to:\[2x + 13 = -24\]It further simplifies to:\[2x = -24 - 13\]So,\[2x = -37\]Therefore,\[x = -37/2\]x equals -18.5.

Step 3: Substitute

Substitute x = -18.5 in the second equation to find the value of y:\[2(-18.5) - y = 12\]This simplifies to:\[-37 - y = 12\]It further simplifies to:\[-y = 12 + 37\]Therefore,\[y = -49\]So, the solution to the system is x = -18.5, y = -49.

Step 4: Check solutions

Checking the solution can be done by graphing or by plugging the solution into the original equation to see if both sides are equal. The solution passes the check if the point lies on both graphs of the equations in the system.

Key Concepts

Elimination Method

The elimination method is a fundamental technique used to solve systems of equations. It's particularly useful when you want to eliminate one of the variables. This approach involves adding or subtracting equations to remove the variable and simplify the system to one equation with one unknown.

To use the elimination method, you follow these steps:

• First, manipulate the equations so that the coefficients of one of the variables are equal, allowing you to eliminate it by addition or subtraction.
• Add or subtract the equations to eliminate one variable. This will give you a single equation with one variable, which you can then solve directly.
• Once you've solved for one variable, substitute this value back into either original equation to find the other variable's value.

In our example, we multiplied the second equation by a factor to align the coefficients of the 'y' variable. This made it possible to eliminate 'y' through subtraction.

Fractions in Equations

Fractions often appear in algebraic equations and can be a bit intimidating, but they're manageable with the right strategies. Simplifying equations with fractions usually involves removing the fractions first to make the equation easier to solve.

Here's how you can handle fractions when solving a system of equations:

• First, identify the least common multiple (LCM) of the denominators to eliminate the fractions.
• Multiply each term in the equation by this LCM to clear the fractions. This transforms the equation into a form that's simpler to work with.
• Continue solving the equation normally once the fractions are removed.

In the provided exercise, multiplying through by the least common multiple of the fractions (in this case, 6) transformed the equation into a simpler form without fractions, facilitating easier manipulation and solution finding.

Graphing Utility

Graphing utilities are powerful tools that help visualize equations and their potential solutions. They can range from graphing calculators to software programs available online.

Here's why using a graphing utility can be beneficial when checking solutions for equations:

• Graphing allows you to see where two or more graphs intersect, which indicates the solutions to the system of equations – the point where the equations are simultaneously true.
• It provides a visual confirmation, ensuring the algebraic solution aligns with the graphical representation.
• Graphing utilities can handle complex equations and plot them quickly, saving time and minimizing human error in manual graphing.

For our problem, a graphing utility could be used to plot both transformed equations. The intersection point would verify if our solution \(x = -18.5, y = -49\) lies on both graphs.

Checking Solutions

Verifying candidate solutions is crucial because computational or algebraic errors can easily occur when solving systems of equations. Checking solutions involves substituting the variables back into the original equations to ensure all equations are satisfied.

Here's a reliable method to check solutions:

• Substitute the values of the identified solutions back into each of the original equations.
• Calculate the left-hand side (LHS) and the right-hand side (RHS) of each equation to ensure they match.
• If both sides of each equation are equal when the solution values are substituted, the solution is correct.

Additionally, leveraging a graphing utility provides an extra layer of checking. By visually confirming that the solution lies at the intersection of both graphs, you reinforce the accuracy of your solution. This dual method of algebraic substitution and graphical verification ensures a comprehensive solution check.