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 Begin…Important
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
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
• Here’s the problem:
Equation one
-x+y=1
Equation two
2x+y=-2
• Try this on your
own…but if you need
help or a few
pointers…see 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
Here’s 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 –1…but 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?
You’re done; simply write out the solution as (-1,0)
***Did you remember??
To write a solution, once you’ve found x and y, you must put x first and then y: (x,y)
Things to Know About the
Substitution Method
• 1. It doesn’t 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 don’t 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
• Here’s an example…try 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
Here’s 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!
What’s left, you may be thinking…well, you have only found “y”…what
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
Here’s 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.
An Example of the Graph and
Check Method
• Here’s the problem:
Equation one
x+y=(-2)
Equation two
2x-3y=(-9)
• Try this problem
out…but a step by step
process follows!
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)
Don’t Let These Fool You…
Special types of Linear Systems
Linear Systems with NO Solution
• Here’s the problem:
Equation one: 2x+y=3
Equation two: 4x+2y=8
• After trying the graph method, you’ll find that the
lines are parallel( don’t 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 <or >.
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 necessary…or
go forward one page to get step by step
guidance.
So, You Needed Help…
• Here’s 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)