In Exercises 1–18, solve each system by the substitution method. xy=6, 2x-y=1

Verified step by step guidance

1. Step 1: Solve one of the equations for one variable. From the equation '2x - y = 1', solve for 'y' to get 'y = 2x - 1'.
2. Step 2: Substitute the expression for 'y' from Step 1 into the other equation. Replace 'y' in the equation 'xy = 6' with '2x - 1' to form a new equation: 'x(2x - 1) = 6'.
3. Step 3: Simplify and solve the equation obtained in Step 2. Expand and rearrange the equation 'x(2x - 1) = 6' to form a quadratic equation '2x^2 - x - 6 = 0'.
4. Step 4: Factorize the quadratic equation or use the quadratic formula to find the values of 'x'.
5. Step 5: Substitute the values of 'x' back into the expression for 'y' obtained in Step 1 ('y = 2x - 1') to find the corresponding values of 'y'.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously. In this case, we have two equations involving the variables x and y, and we need to find their intersection point.

Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable in terms of the other. This expression is then substituted into the other equation, allowing for the determination of the variable's value. This method is particularly useful when one equation is easily solvable for a variable.

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines on a coordinate plane. Each equation in the system represents a line, and the solution to the system is the point where these lines intersect. Understanding the properties of linear equations is essential for solving systems effectively.