Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions algebraically. \(\left\\{\begin{array}{c}x+7 y=12 \\ 3 x-5 y=10\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is x = 5 and y = 1 as the checked solutions validate the original equations.

Step 1: Multiply The Equations

To eliminate one variable, the equations must be manipulated in a way that adding or subtracting them can eliminate that variable. This may require multiplying one or both of the equations by an appropriate number. Multiply the first equation by 3 and the second equation by 1, bringing us to: \[3*(x + 7y) = 3*12 \text{ and } 1*(3x - 5y) = 1*10 \] which simplify to \[3x + 21y = 36 \text{ and } 3x - 5y = 10 \] respectively.

Step 2: Eliminate Variable

Now subtract the second newly formed equation from the first to eliminate the variable x: \[(3x + 21y) - (3x - 5y) = 36 - 10 \] which simplifies to \[26y = 26 \] after subtraction.

Step 3: Solve For Y

Dividing the final equation of step 2 by 26 to isolate y: \[y = 26/26 = 1 \]. This gives the solution for y.

Step 4: Substitute Y and Solve For X

Substitute y = 1 into the first original equation (x + 7y = 12) to solve for x: \[x + 7(1) = 12 \] This simplifies to \[x + 7 = 12 \Rightarrow x = 12 - 7 = 5 \]. So x = 5 is the solution for x.

Step 5: Check The Solutions

Substitute x = 5 and y = 1 into both original equations: \[5 + 7(1) = 12 and 3(5) - 5(1) = 10, \] simplifying to \[12 = 12 and 15 - 5 = 10, \] which satisfy both original equations. Therefore, the solution x = 5, y = 1 is correct.

Key Concepts

Elimination Method

The elimination method is a popular technique for solving systems of equations. The main idea is to remove, or eliminate, one variable, so that you're left with a simpler equation in one variable. Here's how it generally works:

• Choose one of the variables to eliminate. Take a look at the coefficients of the variables in each equation to decide which one might be easiest to eliminate.
• Manipulate the equations as necessary so that adding or subtracting them will eliminate the chosen variable. You might need to multiply one or both equations by suitable numbers to achieve this.
• Add or subtract these modified equations to eliminate one variable, thus simplifying the system to a single equation in one variable.

In this problem, we decided to eliminate the variable \(x\). By multiplying the first equation by 3, we aligned the coefficients with the second equation. Subtracting then gave us an equation with only \(y\), which simplifies the process of finding the solution.

Algebraic Solution

Once a system of equations is reduced to a single variable using elimination, solving it becomes straightforward.

Here's how we approached it in this exercise:

• With the modified equations, we subtracted to get \(26y = 26\). We then solved for \(y\) by dividing both sides by 26, giving us \(y = 1\).
• Having found \(y\), we went back to one of the original equations to solve for \(x\). We substitute \(y = 1\) into \(x + 7y = 12\), leading us to solve for \(x\), resulting in \(x = 5\).

This algebraic process breaks down the problem into manageable steps, making sure each variable is isolated and solved accurately. This approach demonstrates the power and simplicity of algebra when handling systems of equations.

Checking Solutions

After solving a system of equations, it is crucial to verify that your solutions satisfy all original equations. This ensures accuracy in your final answers.

Here’s how to check solutions using the example problem:

• Substitute your found values for \(x\) and \(y\) back into the original system of equations. For this problem, we substituted \(x = 5\) and \(y = 1\) into both equations \(x + 7y = 12\) and \(3x - 5y = 10\).
• Calculate both sides of each equation separately to confirm that they are equal. When we checked: \(5 + 7(1) = 12\) and \(3(5) - 5(1) = 10\), both matched, confirming \(12 = 12\) and \(10 = 10\).

Checking these calculations ensures that the solutions are correct and consistent with the original system of equations. This final step of verification is a crucial part of solving algebraic problems, giving you confidence in your solutions.