3.2 Solving Systems of Equations by algebraically. Main Ideas Solve systems of linear equation by using substitution. Solve systems of linear equations by using elimination.

Systems of Equations Three types of solutions One solution The equations intersect in one point. Infinitely many solutions The equations are the same line. No solutions The equation do not intersect, parallel.

Substitution Method One equation is solved for one variable. Then, this expression is substituted for that variable in the other equation. Hint: It is easier to solve for a variable that has a coefficient of 1. If the variable cancels. Then there are two possible answers ◦ Infinitely many solutions if the numbers are equal. ◦ No solution if the numbers are not equal.

Solve using substitution. 1) 2x – 3y = 22) x + 4y = 26 x + 2y = 15 x – 5y = -10

Solve using substitution. 3) x + y = 1.54) y = 3x – 4 3x + 3y = 4.5 y = 4 + x

Elimination Method Using this method, you eliminate one of the variables by adding or subtracting the equations. To eliminate a variable you must have Same coefficients – add equations Opposite coefficients– subtract equations Sometimes you have to multiply an equation(s) to get the coefficients you need to use elimination. If the variable cancels. Then there are two possible answers ◦ Infinitely many solutions if the numbers are equal. ◦ No solution if the numbers are not equal.

Solve using elimination method. 1) 2x + y = 42) 5b = a 3x + y = 8 2a + 4b = 7

Solve by using elimination. 5) 2x + 3y = 126) x + y = 3 5x – 2y = 11 2x + 2y = 6

Real World problems Keys to solving two variable equations. 1. Read carefully and be organized. 3. Write two equations. 4. Solve the system of equations. 5. Answer the question and label.

Campus Rentals rents 2 and 3 bedroom apartments for $700 and $900 per month, respectively. Last month they had six vacant apartments and reported $4600 in lost rent. How many 2 bedroom apartments were vacant?

Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13,930 for the 48 outdoor chairs sold. How many rocking chairs were sold?