Transcript of Systems of Linear Equations How to: solve by graphing, substitution, linear combinations, and...
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- 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
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- What is a Linear System, Anyways? A linear system includes two,
or more, equations, and each includes two or more variables. linear
system.When two equations are used to model a problem, it is called
a linear system.
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- 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
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- 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!
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- Example: The Substitution Method Heres the problem: Equation
one -x+y=1 Equation two 2x+y=-2 Try this on your ownbut if you need
help or a few pointerssee the next page!
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- 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)
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- 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!)
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- 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.
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- Solving Linear Systems by Linear Combinations
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- 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
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- 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
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- 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!
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- 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
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- A Final way to Solve Systems: Graph and Check
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- 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 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.
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- An Example of the Graph and Check Method Heres the problem:
Equation one x+y=(-2) Equation two 2x-3y=(-9) Try this problem
outbut a step by step process follows!
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- 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)
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- Dont Let These Fool You Special types of Linear Systems
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- 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.
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- 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.
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- Solving Systems of Linear Inequalities
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- 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. 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.
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- An Example of Graphing Linear Inequalities Y1 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.
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- So, You Needed Help Heres the original problem: y 1 First, make
a few basic conclusions: * The line for both boundaries will be
dotted or dashed because it is. *both will be horizontal lines
because there is no x whatsoever in either equation Now, you can
graph the equation (next page)
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- 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
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- In Simpler Terms: Graphing Systems of Linear Inequalities 1.
Sketch the lines of each inequality (remember to use dashed lines
for and solid lines for ) 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.
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- 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.
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- 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
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- 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
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- 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)
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