Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions algebraically. \(\left\\{\begin{array}{l}2 x+3 y=18 \\ 5 x-y=11\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the given system of equations is \(x=3\) and \(y=4\)

Step 1: Transform the equations

Multiply the second equation by 3 to get a matching coefficient with the y term in the first equation. The transformed system of equations will be: \n \(\left\{\begin{array}{l}2 x+3 y=18 \ 15 x-3 y=33\end{array}\right.\)

Step 2: Eliminate a variable

Add the two equations in the system to eliminate y, which results in \(17x = 51\)

Step 3: Solve for x

Divide both sides of the equation \(17x=51\) by 17 to solve for x, which gives: \(x=3\)

Step 4: Substitute to find y

Substitute x=3 into the first equation of the original system (2*3 + 3y = 18), then solve for y. By doing this, you will get: \(y=4\)

Step 5: Check the solution

The solution should satisfy both original equations, so plug \(x=3\) and \(y=4\) into original equations to confirm. For the first equation \(2(3) + 3(4) = 18\) and \( 5(3)-4 = 11\) for the second equation. Indeed, both equations are true.

Key Concepts

Elimination Method

The elimination method is a powerful technique for solving systems of linear equations, by eliminating one variable to find the value of another. It works best when the coefficients of one of the variables are the same or can be made the same.
Here's how it works:

• First, you may need to multiply one or both equations by a number so that the coefficients in front of one variable become the same in both equations.
• Next, choose whether you'll add or subtract the equations. This choice depends on the coefficients. You will "eliminate" one variable when their coefficients sum to zero.
• Once you eliminate the variable, solve for the remaining variable.
• Finally, substitute the found value into one of the original equations to find the other variable.

In our example, we multiplied the second equation by 3, which made the coefficients for "y" in both equations "3" and "-3". By adding these equations, the "y" terms cancel each other out, and we solve for "x" straightforwardly.

Solving Linear Equations

After using the elimination method to remove one variable from the equations, what remains is a linear equation that's much simpler to solve.
A linear equation is an equation of the first degree, meaning its graph is a straight line. These are the steps to solve it:

• Isolate the variable on one side of the equation.
• Use inverse operations, which include addition, subtraction, multiplication, or division, to solve for the variable.

For our system, after eliminating "y", we arrived at the equation 17x = 51. Dividing both sides by 17, we quickly found that x = 3. Solving linear equations is about systematic application of algebraic operations to simplify the problem.

Algebraic Solution Verification

Once you obtain a solution, it's essential to verify its correctness. Algebraic verification is not just a formality; it's crucial for ensuring the validity of your results.
Verification involves plugging the found values back into the original equations to check if they hold true. Here's a simple way to verify:

• Use the values of both variables obtained from the elimination process in the original set of equations.
• Substitute one variable at a time and simplify.
• If both sides of the equation match after substitution for each equation, then your solution is correct.

For the example, substituting x = 3 and y = 4 into both original equations, each side equaled the constant value on the right, confirming the solution's accuracy. Verification is a critical final step to ensure no errors occurred in the elimination and solving process.