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. Let the shortest side of the right triangle be represented by the variable \(x\) meters.
2. Since one of the other sides is 7 meters longer than the shortest side, represent this side as \(x + 7\) meters.
3. The longest side (hypotenuse) is given as 13 meters. Using the Pythagorean theorem for a right triangle, set up the equation: \(x^2 + (x + 7)^2 = 13^2\).
4. Expand the squared term and simplify the equation: \(x^2 + (x^2 + 14x + 49) = 169\).
5. Combine like terms to form a quadratic equation: \$2x^2 + 14x + 49 = 169\(. Then, subtract 169 from both sides to set the equation to zero: \)2x^2 + 14x + 49 - 169 = 0$.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In this problem, two variables represent the unknown side lengths, and their relationships form the system to solve.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. This relationship helps form one equation relating the side lengths, essential for solving the problem.

Algebraic Expression of Word Problems

Translating word problems into algebraic expressions involves defining variables and writing equations based on given relationships. Here, expressing one side as '7 meters longer than the shortest side' creates an equation linking the two unknown sides, enabling the use of systems of equations.