Systems of Linear Equations $$ x \cdot y = 17 \times 7 = 119 $$

Systems of Linear Equations

In this lesson, we study a system of linear equations involving two variables. The goal is not only to find the values of x and y, but also to determine their product based on the given equations.

1. Theory of Linear Equation Systems

A system of linear equations consists of two or more equations with the same variables. By solving the system, we can find values that satisfy all equations at the same time. Common methods include elimination and substitution.

2. Concept Explanation

We are given two equations. One equation represents the sum of two variables, while the other represents their difference. By combining these equations strategically, we can eliminate one variable and solve for the other.

3. Steps or Formula

1. Add the two equations to eliminate one variable.
2. Solve the resulting equation for the remaining variable.
3. Substitute the known value back into one equation.
4. Find the value of the second variable.
5. Multiply the values of x and y.

4. Example Problem

Given the system of equations:

$$ \begin{cases} x + y = 24 \\ x - y = 10 \end{cases} $$

Step 1: Eliminate one variable.

Add both equations to remove y.

$$ (x + y) + (x - y) = 24 + 10 $$ $$ 2x = 34 $$ $$ x = 17 $$

Step 2: Substitute the value of x.

Substitute x = 17 into the first equation.

$$ 17 + y = 24 $$ $$ y = 7 $$

Step 3: Find the product of x and y.

$$ x \cdot y = 17 \times 7 = 119 $$

5. Final Answer

The product of x and y is 119.

Systems of linear equations are useful in real-life situations such as profit sharing, budget planning, and other problems where totals and differences are known.