Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for \$305, while 2 goats and 9 sheep sell for \$285. Find the cost of a single goat and of a single sheep.

Verified step by step guidance

1. Define variables to represent the unknowns: let \(g\) be the cost of one goat and \(s\) be the cost of one sheep.
2. Translate the problem into a system of linear equations based on the given information: \(6g + 5s = 305\) \(2g + 9s = 285\)
3. Choose a method to solve the system (substitution or elimination). For elimination, multiply the equations to align coefficients for either \(g\) or \(s\) so you can eliminate one variable by addition or subtraction.
4. Perform the multiplication and then add or subtract the equations to eliminate one variable, resulting in a single equation with one variable.
5. Solve the single-variable equation for that variable, then substitute back into one of the original equations to find the value of the other variable.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In this problem, the prices of goats and sheep are variables, and the given total costs form the equations.

Setting Up Equations from Word Problems

Translating a word problem into mathematical equations involves identifying quantities and their relationships. Here, the number of goats and sheep and their total costs are used to form equations representing the problem. Accurate translation is essential for solving the system correctly.

Methods for Solving Systems (Substitution or Elimination)

Common methods to solve systems include substitution, where one variable is expressed in terms of another, and elimination, where equations are added or subtracted to eliminate a variable. Applying these methods helps find the individual prices of goats and sheep efficiently.