Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{l} 3 x-5 y=2 \\ 2 x+5 y=13 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \(x = 3\) and \(y = 1.4\). The values \(x = 3\) and \(y = 1.4\) are verified by substituting them back into the original equations, both of which are satisfied.

Step 1: Add the Equations

The two equations in the system are: \(3x - 5y = 2\) and \(2x + 5y = 13\). Because the variable \(y\) has the same coefficient but different signs in these two equations, adding the equations will eliminate the \(y\) variable: \(3x - 5y + 2x + 5y = 2 + 13\) => \(5x = 15\).

Step 2: Solve for the remaining variable

Having eliminated \(y\) by addition, now there's a simple linear equation remaining. Solving this for \(x\) by dividing both sides by 5 gives \(x = 15 / 5\) => \(x = 3\).

Step 3: Substitute \(x\) into one of the original equations

Substitute \(x = 3\) into the equation \(3x - 5y = 2\). This gives \(3*3 - 5y = 2\), or \(9 - 5y = 2\), which simplifies to \(5y = 9 - 2\) => \(5y = 7\). Solving for \(y\) gives \(y = 7 / 5\) => \(y = 1.4\).

Step 4: Check the solutions algebraically

The found solution is \(x = 3\) and \(y = 1.4\). This is checked by substituting these values into the original equations. Substituting into \(3x - 5y = 2\) gives \(3*3 - 5*1.4\), which simplifies to 2, confirming the equality. Substituting into \(2x + 5y = 13\) gives \(2*3 + 5*1.4\), which simplifies to 13, confirming the equality. Thus, the found values for \(x\) and \(y\) indeed are the solution to the system.

Key Concepts

Understanding System of Equations

A system of equations involves a set of two or more equations with multiple variables. Each equation in the system shares a common set of variables, and the goal is to find values for these variables that satisfy all the equations simultaneously.

In our exercise, we are given a system of two linear equations:

• \(3x - 5y = 2\)
• \(2x + 5y = 13\)

The objective is to find values for \(x\) and \(y\) that work for both equations. In such problems, you might use methods like the substitution method, graphing, or—as in this exercise—the elimination method.

Each technique has its own strengths. The elimination method is particularly effective when the equations are easy to manipulate in such a way that one of the variables cancels out.

Linear Equations and the Elimination Method

Linear equations are equations where the variables are raised only to the power of one. They form straight lines when graphed. The equations we are working with are linear because they include linear terms only, like \(3x\) and \(-5y\).

The elimination method, employed here, uses the principle of adding or subtracting equations to eliminate one variable and solve for the other. In this system:

• Identify the variable to eliminate, often chosen based on how their coefficients align.
• Add or subtract the equations to eliminate that variable.
• Solve the resulting single-variable equation.

In our problem, the equations are set up perfectly to eliminate \(y\) since they have coefficients \(5\) and \(-5\) respectively. Adding the equations directly canceled \(y\), simplifying our path to discover \(x\).

The key advantage of using the elimination method is its straightforwardness in scenarios like this, where coefficients are easily compatible, simplifying the system significantly.

Solution Verification

Solution verification, an integral step when solving systems of equations, ensures that the values obtained satisfy all given equations. Without it, there’s a risk of errors or misinterpretations.

After solving for \(x = 3\) and \(y = 1.4\), these values need to be substituted back into both original equations:

• Substituting into \(3x - 5y = 2\) becomes \(3*3 - 5*1.4 = 2\). This checks out perfectly as 9 minus 7 indeed equals 2.
• Substituting into \(2x + 5y = 13\) becomes \(2*3 + 5*1.4 = 13\), confirming that our solution satisfies this equation too.

By checking both equations, we confirm the solution is valid. Always include verification to catch errors and reinforce understanding, ensuring the calculated values are indeed the solution to the system.