Systems of Equations Steps
elimination method example
A method used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations of the system. x − 2y = −19 5x + 2y = 1 Step 1 x − 2y =−19 +5x + 2y = 1 ∴ 6x + 0 =>−18 Step 2 6x = −18 6x ÷ 6 = −18 ÷ 6 x = −3 Step 3 plug x = -3 into x − 2y = −19 now, solve −3 − 2y = −19 for y −3+ 3 − 2y = −19 + 3 −2y = −16 y = 8 Step 4 solution (−3,8)
system of linear equations
A set of two or more linear equations containing two or more variables. 2x + 3y = −1 x − 3y = 4
Graphing a system of equations-Steps
For both equations 1. Put into y=mx+b format 2. make a table by selecting 3 values for x and solving for y. 3. Plot the point from the table. 4. Draw the line Repeat for 2nd line. The solution is the point where the two lines intersect.
System of equations with no solution
The lines are parallel. The equations share the same slope, but different y-intercepts. y = 2x +8 y = 2x +12
System of equation with exactly one solution
The systems have different slopes and can be solved. x + y = 7 x − y = 1 Solution: (4, 3)
Link to Graphing with Rise over Run
http://www.wikihow.com/Graph-Linear-Equations
How does the graph of two parallel lines look?
parallel lines inconsistent 0 solutions same slope, different intercepts
Methods for solving a system of equations
1) graphing 2) substitution 3) elimination
Substitution Method-Steps
A method of solving a system of equations when you solve one equation for a variable, substitute that expression into the other equation and solve, and then use the value of that variable to find the value of the other variable. Step 1 Choose one equation and solve for x or y. Step 2 Substitute the expression from that equation into the other equation and solve. Step 3 Substitute the value found in step 2 back into the equation solved step one.
Elimination method-Steps
A method of solving systems of equations where you put both equations in standard form, add or subtract the equations to eliminate a variable and solve for the variable left, and then use that number to find the value of the other variable. Steps: 1. Target one variable for elimination 2. Multiply one (or both) of the equations so that targeted variables are opposites. Then, rewrite the new system. 3. Add the two equations and write the resulting sum. (You should only have one variable left.) 4. Solve for the remaining variable. 5. Substitute the value of the variable found in step 3 into one of the original equations and solve for the other variable. 6. Check your solution in both original equations.
substitution method example
A method used to solve systems of equations by solving an equation for one variable and substituting the resulting expression into the other equation(s). y = 2x y = x + 5 Step 1 y = 2x y = x + 5 Step 2 plug 2x into y = x + 5 for y 2x = x + 5 Step 3 solve for x 2x − x = x − x + 5 x = 5 Step 4 plug x = 5 into y = 2x y = 2(5) y = 10 Step 5 solution (5, 10)
system of linear inequalities
A set of two or more linear inequalities containing two or more variables. 2x + 3y > −1 x − 3y ≤ 4
solution of a system of linear equations
Any ordered pair that satisfies all the equations in the system. x + y = −1 −x + y = −3 Solution: (1, −2)
System of equations with infinitely many solutions
When simplified, the two lines are the same. Both equations have the same slope and same y-intercept. x + y = 2 2x + 2 y = 4
How does a the graph of two intersecting lines look?
intersecting lines consistent & independent 1 solution different slopes