Solve each problem using a system of equations in two variables. See Example 6. The longest side of a right triangle is 13 m in length. One of the other sides is 7 m longer than the shortest side. Find the lengths of the two shorter sides of the triangle.

Verified step by step guidance

1. Identify the variables: Let x be the length of the shortest side, and y be the length of the other side.
2. Set up the first equation using the Pythagorean theorem: x^2 + y^2 = 13^2.
3. Set up the second equation based on the problem statement: y = x + 7.
4. Substitute the expression for y from the second equation into the first equation: x^2 + (x + 7)^2 = 13^2.
5. Simplify and solve the resulting quadratic equation to find the value of x, then use y = x + 7 to find the value of y.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. To solve such a system, one must find the values of the variables that satisfy all equations simultaneously. This can be done using various methods, including substitution, elimination, or graphing. In this problem, we will set up equations based on the relationships described in the triangle.

Right Triangle Properties

A right triangle is defined by having one angle equal to 90 degrees. The sides of a right triangle are related by the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Understanding these properties is crucial for setting up the equations needed to solve the problem.

Variable Representation

In algebra, variables are symbols used to represent unknown values. In this problem, we can represent the lengths of the two shorter sides of the triangle with variables, such as 'x' for the shortest side and 'y' for the longer side. Properly defining these variables based on the problem's context allows us to create equations that can be solved to find the actual lengths of the sides.