Bell Ringer 2 Systems of Equations 4 A

Bell Ringer 2.

Systems of Equations 4 A system of equations is a collection of two or more equations with a same set of unknowns A system of linear equations can be solved four different ways: Substitution Elimination Matrices Graphing

Review of Terms ● A variable is a letter which represents an unknown number. Any letter can be used as a variable. ● An algebraic expression contains at least one variable. Examples: a, x+5, 3 y – 2 z ● A verbal expression uses words to translate algebraic expressions. Example: “The sum of a number and 3” represents “n+3. ”

● An equation is a sentence that states that two mathematical expressions are equal. Example: 2 x-16=18

Steps to Solving Equations ● Simplify each side of the equation, if needed, by distributing or combining like terms. ● Move variables to one side of the equation by using the opposite operation of addition or subtraction. ● Isolate the variable by applying the opposite operation to each side. • First, use the opposite operation of addition or subtraction. • Second, use the opposite operation of multiplication or division. ● Check your answer.

Examples ● “y” is the variable. ● Add 6 to each side to isolate the variable. ● Now divide both sides by 3. ● The answer is 5. ● Check the answer by substituting it into the original equation.

Try this. . . Did you get x = - 4? You were right!

Solving Systems of Equations using Substitution Steps: 1. Solve one equation for one variable (y= ; x= ; a=) 2. Substitute the expression from step one into the other equation. 3. Simplify and solve the equation. 4. Substitute back into either original equation to find the value of the other variable. 5. Check the solution in both equations of the system.

Example #1: y = 4 x 3 x + y = -21 Step 1: Solve one equation for one variable. y = 4 x (This equation is already solved for y. ) Step 2: Substitute the expression from step one into the other equation. 3 x + y = -21 3 x + 4 x = -21 Step 3: Simplify and solve the equation. 7 x = -21 x = -3

y = 4 x 3 x + y = -21 Step 4: Substitute back into either original equation to find the value of the other variable. 3 x + y = -21 3(-3) + y = -21 -9 + y = -21 y = -12 Solution to the system is (-3, -12).

y = 4 x 3 x + y = -21 Step 5: Check the solution in both equations. Solution to the system is (-3, -12). y = 4 x -12 = 4(-3) -12 = -12 3 x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21= -21

Example #2: x + y = 10 5 x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5 x - y = 2 5 x -(-x +10) = 2

x + y = 10 5 x – y = 2 Step 3: Simplify and solve the equation. 5 x -(-x + 10) = 2 5 x + x -10 = 2 6 x = 12 x=2

x + y = 10 5 x – y = 2 Step 4: Substitute back into either original equation to find the value of the other variable. x + y = 10 2 + y = 10 y=8 Solution to the system is (2, 8).

x + y = 10 5 x – y = 2 Step 5: Check the solution in both equations. Solution to the system is (2, 8). x + y =10 2 + 8 =10 10 =10 5 x – y = 2 5(2) - (8) = 2 10 – 8 = 2 2=2

Solve by substitution: 1. 2.