Systems of Linear Equations How to: solve by graphing, substitution, linear combinations, and special types of linear systems By: 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 know Linear system: two equations that form one equation Solution: 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 Systems By: Substitution Solving Linear Systems by Substitution Basic steps: 1. Solve one equation for one of its variables 2. Substitute that expression into the other equation and
solve for the other variable 3. Substitute that value into first equation; solve 4. Check the solution See next page for a step by step example! Example: The Substitution Method Heres the problem: Equation one -x+y=1 Try this on your Equation two 2x+y=-2
ownbut if you need help or a few pointerssee the next page! First, solve equation one for y Y=x+1 Next, substitute the above expression in for y in equation two, and solve for x Heres 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=-1 Congratulations! 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=0 So, 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 Method 1. 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 Solve Here is an instance where it is easier to solve equation two first (for x)
Equation One: 3x-2y=1 Equation Two: x+4y=3 By 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 x Here is an instance where you help yourself by solving equation one for y Equation One: 2x+y=5 Equation Two: 3x-2y=11
You 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 Combinations Basic steps: 1. Arrange the
equations with like terms in columns 2. 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 variable 4. Substitute the value gotten in step 3 into either of the original equations; solve for other variable 5. Check the solution in both original equations Example: Solving Systems by Linear Combinations
Heres an exampletry it out, but if you have any problems, see the next page for a guided, step by step explanation Solve this linear equation: Equation One: 3x+5y=6 Equation Two: -4x+2y=5 Heres the original problem: Solve the linear system Equation 1: 3x+5y=6 Equation 2: -4x+2y=5
Do you remember the first step? put the equations into columns 3x+5y=6 -4x+2y=5 Now, 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 4 3x+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=15 Your 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=5 Substitute the value you just found for y : 3
2 -4x+2(3)=5 2 simplify by multiplying 2 by three-halves -4x+3=5 subtract 3 from both sides because you are working to isolate x -4x=2 solve for x by dividing both sides by 4 (inverse operation) x=-1 2 The 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 check Basic steps: 1. Put each equation into slope intercept form (y=Mx+B) 2. Graph the two
lines (M is your slope; B is your Yintercept) 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 Method Heres Try thisthe problem problem:
outbut Equation a stepone by step x+y=(2) Equation process follows!two 2x-3y=(-9) The first step is to put the equations into slope-intercept form Equation one: originally, it was: x+y=(-2) but after putting it into slope intercept, it reads: y=(-x)-2 Equation two: originally, it was: 2x-3y=(-9), but once in slope intercept, it reads:
y= 2x+3 3 From the above equations, you can make the following conclusions: Equation one has a slope of 1 and a y intercept of 2 Equation 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 intercept now, 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)=-2 Equation two: 2(-3)-3(1)=-6-3=-9 Since 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 You Special types of Linear Systems Linear Systems with NO Solution Heres the problem: Equation one: 2x+y=3
Equation two: 4x+2y=8 After trying the graph method, youll find that the lines are parallel( dont intersect) and therefore have no solution After 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
Solutions If 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 Inequalities Here 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 Inequalities Y<4 Y>1 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 Help Heres the original problem: y<4, y>1
First, 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 equation Now, 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 axis Label the lines Pick points and follow the previously described process to find where to shade (between y=1 and y=4) Write the equation on the line In Simpler Terms: Graphing Systems of Linear Inequalities 1. 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 equation 3. 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 Easier Basic 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 system 2. Using substitution or linear combinations will allow you to get an exact and more accurate answer 3. 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=6 Equation two: 2x+y=10 2. Solve the following Linear system by means of substitution Equation one: x-6y=-19 Equation two: 3x-2y=-9 3. Solve the following Linear system by means of substitution Equation one: x+3y=7 Equation two: 4x-7y=-10 See next page for more; answers on last page
A Little More Fun: More Examples 4. Use linear combinations to solve this system Equation one: -2x-3y=4 Equation two: 2x-4y=3 5. Use linear combinations to solve this system Equation one: 3x-5y=-4 Equation two: -9x+7y=8 Answers to the Examples
1. 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) The End
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