TODAY: MAKE UP TESTS?**KHAN ACADEMY DUE TOMORROW BEGIN SYSTEMS OF EQUATIONS
WARM-UP:1.2. What is the percent change in the price ofgas from $4.80/gal to $5.00/gal.
SYSTEMS OF LINEAR EQUATIONSSo far, we have solved equations with one 3x + 5 = 35variable:two and in variables. That will changeIn both cases, now as we solvethough we have only multiple equationsbeen able to solve at the same time,one equation at a looking for antime. ordered pair which A system of linear equations is solves each simply two or more linear equation, and thus equations using the same is a solution for variables. both.Well start with systems of two equations using twovariables,then increase this to three equations and variables.
WHAT IS A SYSTEM OF LINEAR EQUATIONS? If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x , y) where x and y make both equations true at the same time. If the lines are parallel, there will be no solutions. Ifthe equations are the same line, there will be an infinite number of solutions. There are several methods of solving systems of equations; well look at a couple today.
Tell whether the ordered pair is a solution of the given system. Substitute 5 for x and 2 for y in each (5, 2); equation in the 3x – y = 13 system. 3x – y 13 0 3(5) – 2 13 2–2 0 15 – 2 13 0 0 13 13 The ordered pair (5, 2) makes both equations true. (5, 2) is the solution of the system.
Helpful Hint If an ordered pair does not satisfy the first equation in the system, there is no reason to check the other equations. x + 3y = 4 Substitute –2 for (–2, 2); –x + y = 2 x and 2 for y in each equation in x + 3y = 4 –x + y = 2 the system.–2 + 3(2) 4 –(–2) + 2 2 –2 + 6 4 4 2 4 4 The ordered pair (–2, 2) makes one equation truebut not the other. (–2, 2) is not a solution of the system.
SOLVING LINEAR SYSTEMS BY GRAPHING Consider the following system: x – y = –1 y x + 2y = 5 Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they x (1 , 2)Notice that line. of these points will lie on the anymake the second equation true.However, there is ONE pointthat makes both true at thesame time… The point where they intersect makes both equations true at the same time.
Practice 1Graph the system of equations. Determinewhether the system has one solution, nosolution, or infinitely many solutions. If thesystem has one solution, determine the solution.
Practice 1 y The two equations in slope-intercept form are: x Plot points for each line. Draw the lines.These two equations represent the same line.Therefore, this system of equations has infinitely many solutions .
Practice 2 y The two equations in slope- intercept form are: x Plot points for each line. in the lines. DrawThis system of equations representstwo parallel lines.This system of equations has no solution because these twolines have no points in common.
Practice 3 y The two equations in slope-intercept form are: x Plot points for each line. Draw in the lines.This system of equations represents two intersecting lines.The solution to this system of equations is a single point (3,0)
Graphing to Solve a Linear System Lets summarize! There are 4 steps to solving a linear system using a graph.Step 1: Put both equations in slope - Solve both equations for y, so thatintercept form. each equation looks like y = mx + b.Step 2: Graph both equations on the Use the slope and y - intercept forsame coordinate plane. each equation in step 1.Step 3: Estimate where the graphs This is the solution! LABEL theintersect. solution!Step 4: Check to make sure your Substitute the x and y values into bothsolution makes both equations true. equations to verify the point is a solution to both equations.
Like variables Solve: must be lined under each by ELIMINATION other. 5x - 4y = -21 -2x + 4y = 18We need toeliminate 3x = -3 Divide by 3(get rid of)a variable. x = -1The y’s bewill theeasiest.So,we will add THEN----the twoequations.
5x - 4y = -21 5(-1) – 4y = -21Substituteyour answer -5 – 4y = -21into either 5 5originalequation and -4y = -16solve for thesecondvariable. y=4 Answer (-1, 4) Now check our answers in both equations------
Like variables Solve: must be lined under each by ELIMINATION other. 2x + 7y = 31 5x - 7y = - 45We need toeliminate 7x = -14 Divide by 7(get rid of)a variable. x = -2The y’s willbe theeasiest. So,we will add THEN----the twoequations.
Substituteyour answer 2X + 7Y = 31into eitheroriginal 2(-2) + 7y = 31equation and -4 + 7y = 31solve for thesecond 4 4variable. 7y = 35 y=5 Answer (-2, 5) Now check our answers in both equations------
Like variables must be lined under each Solve: other. by ELIMINATION x + y = 30We need to eliminate(get rid of) a variable. x + 7y = 6To simply add thistime will not eliminatea variable. If one of thex’s was negative, itwould be eliminatedwhen we add. So wewill multiply oneequation by a – 1.
X + Y = 30 X + Y = 30( X + 7Y = 6 ) -1 -X – 7Y = - 6 Now add the two equations and -6Y = 24 solve. -6 -6 Y=-4 THEN----
Substituteyour answer X + Y = 30into eitheroriginal X + - 4 =30equation andsolve for the 4 4secondvariable. X = 34 Answer (34, - 4) Now check our answers in both equations------
Solve: Elimination By Multiplying x + +y0y 4 4 0x = = Like variables must be lined 2x + 3y = 9 under each other. We need to eliminate (get rid of) a variable.To simply add this time will not eliminate a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. Sowe will multiply the 1st equation by a – 2.
( X + Y = 4 ) -2 -2X - 2 Y = - 8 2X + 3Y = 9 2X + 3Y = 9 Now add the two equations and Y=1 solve. THEN----
Substitute your answer into either original equation and solve for the second variable. X+Y=4 X +1=4 - 1 -1 X=3 Answer (3,1)Now check our answers in both equations--
REVIEW: SOLVING BY GRAPHINGIf the lines cross once, there will be onesolution.If the lines are parallel, there will be nosolutions.If the lines are the same, there will be aninfinite number of solutions.