Systems of Linear Equations

Systems of Linear EquationsHow to: solve by graphing, substitution, linear combinations, and special types of linear systemsBy: Sarah R.Algebra 1; E block

What is a Linear System, Anyways?A linear system includes two, or more, equations, and each includes two or more variables.When two equations are used to model a problem, it is called a linear system.

Before You BeginImportant Terms to knowLinear system: two equations that form one equationSolution: the answer to a system of linear equation; must satisfy both equations***: a solution is written as an ordered pair: (x,y)Leading Coefficient: any given number that is before any given variable (for example, the leading coefficient in 3x is 3.)Isolate: to get alone

Ways to Solve Linear SystemsBy: Substitution

Solving Linear Systems by SubstitutionBasic steps:1. Solve one equation for one of its variables2. Substitute that expression into the other equation and solve for the other variable3. Substitute that value into first equation; solve4. Check the solution See next page for a step by step example!Example: The Substitution MethodHeres the problem:Equation one-x+y=1Equation two2x+y=-2

Try this on your ownbut if you need help or a few pointerssee the next page!First, solve equation one for yY=x+1Next, substitute the above expression in for y in equation two, and solve for xHeres how: Equation two 2x+y=-2 Substitute x+1 for y 2x+ (x+1)=-2 simplify the above expression 3x+1=-2 Subtract one from both sides (because your goal is to solve for x) 3x=-3 Solve for x ( divide both sides by 3; since x is being multiplied by three, and you need it alone, so do the inverse operation: divide by 3) X=-1Congratulations! You now know x has a value of 1but you still need to find y.To do so First, write down equation one Y=x+1 Substitute 1 for x, since you just found that x=-1 Y= (-1)+1 Solve the equation for y by adding 1 +1 Y=0So, now what? Youre done; simply write out the solution as (-1,0)***Did you remember??To write a solution, once youve found x and y, you must put x first and then y: (x,y)

Things to Know About the Substitution Method1. It doesnt matter if you choose to solve for y or x first; the answer or solution will be the same either way.2. You can also choose to solve equation two before equation one; simply follow the same steps, just using a slightly rearranged order. *** You should always decide whether to solve x or y first, or equation one or two first, depending on which way is more efficient (See next page!)Deciding the Order in Which to SolveHere is an instance where it is easier to solve equation two first (for x)Equation One: 3x-2y=1Equation Two: x+4y=3By solving equation two first, you are lessening your work, because there is no leading coefficient before the x in equation two, so you dont have to worry about dividing to isolate the xHere is an instance where you help yourself by solving equation one for y Equation One: 2x+y=5Equation Two: 3x-2y=11You should solve for y in the first equation. Again, you lessen your work because there is no leading coefficient before the y in equation one, while there are leading coefficients with all the other variables.

Solving Linear Systems by Linear Combinations

Solving Systems by means of Linear CombinationsBasic steps:1. Arrange the equations with like terms in columns2. After looking at the coefficients of x and y, you need to multiply one or both equations by a number that will give you new coefficients for x or y that are opposites.3. Add the equations and solve for the unknown variable4. Substitute the value gotten in step 3 into either of the original equations; solve for other variable5. Check the solution in both original equationsExample: Solving Systems by Linear CombinationsHeres an exampletry it out, but if you have any problems, see the next page for a guided, step by step explanationSolve this linear equation:Equation One: 3x+5y=6Equation Two: -4x+2y=5Heres the original problem:Solve the linear systemEquation 1: 3x+5y=6Equation 2: -4x+2y=5Do you remember the first step?put the equations into columns3x+5y=6-4x+2y=5Now, you need to multiply each equation by a number that will cause your leading coefficients of either x or y to become opposites. In this case, try to get opposite coefficients for x. to do this, multiply the first equation by four and the second by three. ***You must multiply all terms by 3 or 43x+5y=6, when all terms are multiplied by four, this equation will be: 12x+20y=24-4x+2y=5, when all terms are multiplied by three, this equation will be: -12x +6y=15Your next step is to add the two revised equations: 12x+ 20y=24+ (-12x) + 6y= 15 26y=39 (sum of equations)To get the y alone, you must divide each side by 26, (you divide since the y is being multiplied by 26, and to isolate the y you do the inverse operation)So, you have found Y, but you aren't done yet!Whats left, you may be thinkingwell, you have only found ywhat about x?To find x, you have to place y into equation 2.Equation 2: -4x+2y=5Substitute the value you just found for y : 3 2-4x+2(3)=5 2simplify by multiplying 2 by three-halves-4x+3=5subtract 3 from both sides because you are working to isolate x-4x=2solve for x by dividing both sides by 4 (inverse operation)x=-1 2The solution to the example system is (-1, 3) 2 2

A Final way to Solve Systems:Graph and Check

Heres a method called graph and checkBasic steps: 1. Put each equation into slope intercept form (y=Mx+B)2. Graph the two lines (M is your slope; B is your Y-intercept)3. Find the point that the lines appear to intersect at, and then put that solution into EACH equation and solve to check for accuracy. An Example of the Graph and Check MethodHeres the problem: Equation onex+y=(-2)Equation two2x-3y=(-9)

Try this problem outbut a step by step process follows!The first step is to put the equations into slope-intercept formEquation one: originally, it was: x+y=(-2) but after putting it into slope intercept, it reads: y=(-x)-2Equation two: originally, it was: 2x-3y=(-9), but once in slope intercept, it reads: y= 2x+3 3From the above equations, you can make the following conclusions:Equation one has a slope of 1 and a y intercept of 2Equation two has a slope of 2 and a y intercept of 3 3***Remember that in the slope intercept form (y=mx+b), m is the slope; b is the y interceptnow, you will be able to graph the two equations as lines. Once done this, you can conclude that the lines seem to intercept at (-3,1).To check this assumption, put (-3) in for x and 1 in for y in BOTH EQUATIONS, and solve both:Equation one: (-3)+(-1)=-2Equation two: 2(-3)-3(1)=-6-3=-9Since both equations, once solved, equaled what they should have, you know that the solution to this linear system is (-3,1)

Dont Let These Fool YouSpecial types of Linear Systems

Linear Systems with NO SolutionHeres the problem:Equation one: 2x+y=3Equation two: 4x+2y=8After trying the graph method, youll find that the lines are parallel( dont intersect) and therefore have no solutionAfter trying either of the substitution or linear combination methods, you will have an equation that cannot be dealt with. You will know that this is the case because it will make no senses whatsoever. Therefore, you have no solution to the system.Linear System with MANY SolutionsIf you use the graph method, you will see that the equations are the same line, and any point on the line is a solution. If you use linear combinations or substitution, you will have a number =number, but both numbers will be the same. For example, 7=7 or 1=1. This indicates that the systems has many solutions. Solving Systems of Linear Inequalities

Graphing Systems of Linear InequalitiesHere are some pointers and things to know:1. The boundary line on the graph will be dashed if the inequality is < or >. 2. The boundary line will be solid if the inequality is . 3. You will also notice that graphs of linear inequalities are shaded in certain areas. To decide where to shade, pick a point that is CLEARLY above the line, and a point that is CLEARLY below the line. Put the first point into the inequality; solve; then do the same for the other point. Whichever point works, you shade that side. An Example of Graphing Linear InequalitiesY1 Try this one out! Remember the steps; you can always go back a page if necessaryor go forward one page to get step by step guidance.So, You Needed HelpHeres the original problem: y1First, make a few basic conclusions:* The line for both boundaries will be dotted or dashed because it is < or >.*both will be horizontal lines because there is no x whatsoever in either equationNow, you can graph the equation (next page)

Graphing Errors

If your graph looked like the previous slide, you can congratulate yourself on getting the lines drawn correctly. However you forgot to:Label the axisLabel the linesPick points and follow the previously described process to find where to shade (between y=1 and y=4)Write the equation on the lineIn Simpler Terms: Graphing Systems of Linear Inequalities1. Sketch the lines of each inequality (remember to use dashed lines for < or > and solid lines for < or >)2. LIGHTLY SHADE the area that is found by choosing points and placing them into the equation3. The final result, or answer, is the area that is where the shaded planes intersect, for example, in the previous problem, the answer is anywhere between the boundary lines of y=4 and y=1.To Make it Somewhat EasierBasic guidelines for linear systems:1. Use the graphing method to get an approximate answer, to check a solution, or to give a visual idea of the system2. Using substitution or linear combinations will allow you to get an exact and more accurate answer3. Substitution helps a lot when used in systems that have coefficients of 1 or 1.4. When there isn't a 1 or 1 as coefficients, the linear combinations method is efficient. Fun, Fun: Examples to do on Your Own(Answers are on Last Page)1. Solve the following Linear System by graphing Equation one: -2x+3y=6Equation two: 2x+y=102. Solve the following Linear system by means of substitutionEquation one: x-6y=-19Equation two: 3x-2y=-93. Solve the following Linear system by means of substitutionEquation one: x+3y=7Equation two: 4x-7y=-10See next page for more; answers on last page A Little More Fun: More Examples4. Use linear combinations to solve this systemEquation one: -2x-3y=4Equation two: 2x-4y=35. Use linear combinations to solve this systemEquation one: 3x-5y=-4Equation two: -9x+7y=8Answers to the Examples1. Your graph should show a point of intersection, which is your solution, of (3,4).2. (-1,3)3. (1,2)4. (-1.5,9)5. (-.5, .5)

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