10.2 The Substitution Method 10.1 Systems of Equations 10.2 The Substitution Method 1/21

10.1 Systems of Equations

-a SYSTEM OF EQUATIONS is a group of equations that you deal with all together at once. -if you have two variable and you need to find a numerical value for both of them, the only way you can do it is if you have multiple equations. If you have to solve for 2 variables you will need 2 equations, if you need 3 variable you will need 3 equations and so on -Initially we will be graphing both equations (just like what we did in chapter 9) but now we will do it with 2 lines on the same graph paper.

SYSTEM SOLUTIONS A system of 2 equations can have three type of solutions. -no solution (lines do not intersect) -one solution (lines intersect once) -infinite solution (lines overlap, coincide, same line) How many solutions you have means that that is how many ordered pairs satisfy BOTH equations.

Terminology Dependent Independent Consistent – The lines intersect or coincide. There are two types of consistent systems. Dependent Independent

Terminology. Inconsistent – means that that lines do not intersect (must be parallel, remember same slope?)

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs of and are parallel, there are no solutions.

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs of and are intersecting lines, there is one solution.

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions. Answer: Since the graphs of and coincide, there are infinitely many solutions.

Graph the system of equations Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. a. Answer: one; (0, 3)

How do you tell how many solutions based on the equation? If the slopes are different then the lines intersect once If the slopes are the same but the y-intercepts are different then the lines do not intersect If the slopes are the same and the y-intercepts are the same then the lines are the same line, so they intersect always, or infinitely.

10.2 The Substitution Method

Many times we may not be able to look at the graph and find the intersection point due to the fact that maybe they intersect somewhere other than the crossing point of two grid lines (we would have to guess) and that is not the best way to get answers. So we must revert to an algebraic technique. We will learn 2 techniques. The first of which is called the SUBSTITUTION METHOD.

Steps for solving SYSTEMS using SUBSTITUTION When using the SUBSTITUTION METHOD we will want either one or both of our equations to be solved for a particular variable. That means that either x or y is all by itself on one side of the equation with a coefficient of 1. Steps for solving SYSTEMS using SUBSTITUTION Solve either equation for x or y. (this technique is really nice to use if this is already done when they present you with the problem) Substitute (replace) the expression that the variable in step 1is equal to into the other equation (this will help remove a variable). Solve for the leftover variable. Take your solution found in step 2 and plug it back into one of the original equations (either one) and solve for the other variable. Now that you have an x and a y value, state your answer as an ordered pair. You may check you solution by plugging the ordered pair back into both equations and know that you are correct if you get 2 true statements.

Solve using substitution Which equation already has a variable solved for?

This system does not have a variable solved for, however we can be intelligent and look at the two equations and determine that a particular variable in one of the equations can be solved for rather quickly, which equation and which variable?

Occasionally both equations will already be solved for the exact same variable.

Something unique happens when we solve this system.

Something unique happens when we solve this system.