Pre Algebra Solving Systems of Equations

Solving Systems of Equations Solving a System of Equations by Graphing ● A system of linear equations contains two or more equations. The solution of such a system is an ordered pair (solution to both equations). ● To solve a system of linear equations, graph both lines on the same coordinate plane, and the point of intersection between the two lines is the solution. ● Steps on Solving a System of Linear Equations using the Graphing Method: 1. Graph and label one equation on the coordinate plane. 2. Graph and label the other equation on the same coordinate plane. 3. The point where the lines intersect is the solution. Write the coordinates on the graph. 4. CHECK the solution in both equations algebraically. The solution must satisfy both equations. ● Ex. y = -2x + 1 and y - 4 = x 1. Rearrange the 2nd equation to isolate y → y = x + 4 2. Graph both equations 3. 4. Find the coordinates for the point of intersection of the two lines Systems of Equations with One solution, No solution, or Infinitely many Solutions ● If lines intersect once, there is one solution (an ordered pair) to the system ● If the lines are parallel, there is no solution to the system because the lines will never intersect. The slope is the same of both lines ● If the lines are the same, there are infinitely many solutions to the system. ○ Lines are the same if they have the same equation when written in slope-intercept form. Applied Systems of Equations using Graphing ● You can model and solve many real-world problems in which you are given two different relationships between two unknown quantities by using systems of linear equations. ○ Define the variables (x,y) ■ Let x = __ ; Let y = __ ○ Write 2 linear equations using the information given in the problem ■ Rewrite the equations in slope-intercept form, if needed. ○ Graph the equations on the same axes. State the slope and y-intercept, and label each line with its equation ○ Write the coordinates of the solution. Interpret the solution based on the problem Solving a System of Equations by Substitution ● Solve one of the equations for either x or y (isolate one of the variables) ● Substitute the expression from step 1 into the other equation in place of the isolated variable. Now the second equation only has one variable. ● Solve that second equation to find the value of that one variable. ● Take this value and substitute it into one of the original equations to find the value of the second variable. ● Write the solution as an ordered pair and check solution Solving a System of Equations using Elimination ● If the two equations are in standard form (Ax+By=C), and each has coefficients that are additive inverses for one of the variables, the equations can be added together by combining like terms to eliminate the variable. ● Then substitute that value into the original equations to find the value of the other variable. State your solution by writing your solution as an ordered pair (unless there is no solution or infinitely many solutions). Solving a System of Equations by Elimination with Multiplication 1. Write both equations in standard form 2. Compare the coefficients of the x-terms. a. Ask yourself, “What number can I multiply one of the equations by to get the additive inverse of the other term?” 3. Write an equation equivalent to the second equation by multiplying each term by the number you found above. 4. Now solve by elimination → Add the equations together and solve for one variable. 5. Substitute the value you found for y into either one of the original equations.