Transcript Topic 5B PowerPoint Student

Systems of
Equations &
Inequalities
Mrs. Daniel
Algebra 1
Table of Contents
1.
2.
3.
4.
5.
6.
What is a System?
Solving Systems Using Substitution
Solving Systems Using Elimination
Solving Systems Using Graphing
Systems & Word Problems
Solving Systems of Inequalities by Graphing
What is a System?!
• System of Linear Equations - a set of two or more
linear equations containing two or more
variables.
– Example: y  x 3

 y 2 x 1
• Solution of a System of Linear Equations – is an
ordered pair that satisfies each equation in the
system.
Three Methods for Solving
Systems
• Graphing
• Substitution
• Elimination
Method selected
depends on information
provided and personal
preference.
Possible Types of Solutions
• One point of intersection, so one solution.
• Parallel lines, so no solution.
• Same lines, so infinite # of solutions.
Solutions
We check to determine if a solution is valid by
plugging in x or y to EACH equation.
Must satisfy both equations to be a valid
solution.
Helpful Hint
If an ordered pair does not satisfy the first equation
in the system, there is no reason to check the other
equations.
Let’s Practice….
Tell whether the ordered pair is a solution of the given system.
(–2, 2);
x + 3y = 4
–x + y = 2
Let’s Practice….
Tell whether the ordered pair is a solution of the given system.
(1, 3);
2x + y = 5
–2x + y = 1
Let’s Practice….
Tell whether the ordered pair is a solution of the given system.
(2, –1);
x – 2y = 4
3x + y = 6
Writing a System of
Equations
Wren and Jenni are reading the same book. Wren is on
page 14 and reads 2 pages every night. Jenni is on page 6
and reads 3 pages every night. After how many nights
will they have read the same number of pages? How
many pages will that be?
Let’s Practice…
Video club A charges $10 for membership and $3
per movie rental. Video club B charges $15 for
membership and $2 per movie rental. For how
many movie rentals will the cost be the same at
both video clubs? What is that cost?
Solving Systems
Using
Substitution
Theory of Substitution Method
• Substitution is used to
reduce the system to
one equation that has
only one variable.
• Then you can solve this
equation for the one
variable and substitute
again to find the other
variable.
How to: Substitution
Method
Solving Systems of Equations by Substitution
Step 1
Step 2
Solve for one variable in at least one equation, if
necessary.
Substitute the resulting expression into the other
equation.
Step 3
Solve that equation to get the value of the first
variable.
Step 4
Substitute that value into one of the original equations
and solve for the other variable.
Step 5
Write the values from steps 3 and 4 as an ordered pair,
(x, y), and check.
Example: Substitution
Solve the system by substitution.
y = 3x
y=x–2
Let’s Practice…
Solve the system by substitution.
y=x+1
4x + y = 6
Let’s Practice…
Solve the system by substitution.
x + 2y = –1
x–y=5
Solving Systems
by Elimination
Theory of Elimination
Method
Since –2y and 2y have opposite coefficients, the y-term is
eliminated. The result is one equation that has only one
variable: 6x = –18.
How to: Elimination
Method
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
Standard Form: Ax + By = C
Look for variables that have the
same coefficient.
Solve for the variable.
Step 4: Plug back in to find the
other variable.
Substitute the value of the variable
into the equation.
Step 5: Check your solution.
Substitute your ordered pair into
BOTH equations.
Let’s Practice…
x+y=5
3x – y = 7
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
They already are!
The y’s have the same
coefficient.
Add to eliminate y.
x+ y=5
(+) 3x – y = 7
Let’s Practice…
x+y=5
3x – y = 7
Step 4: Plug back in to find the
other variable.
Step 5: Check your solution.
Let’s Practice…
Solve
y + 3x = –2
by elimination.
2y – 3x = 14
Let’s Practice…
Solve
3x – 4y = 10
by elimination.
x + 4y = –2
Example: Elimination by Subtracting
4x + y = 7
4x – 2y = -2
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
They already are!
The x’s have the same
coefficient.
Subtract to eliminate x.
4x + y = 7
(-) 4x – 2y = -2
Example: Elimination by Subtracting
4x + y = 7
4x – 2y = -2
Step 4: Plug back in to find the
other variable.
Step 5: Check your solution.
Let’s Practice…
Solve
3x + 3y = 15
by elimination.
–2x + 3y = –5
Let’s Practice…
Solve
2x + y = –5
by elimination.
2x – 5y = 13
When Elimination Is Not Obvious
You can use the elimination method, even if
the coefficients are not the same.
Just multiply all terms in one equation by
the LCM to make the same.
More Elimination
2x + 2y = 6
3x – y = 5
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
They already are!
None of the coefficients are the
same!
Find the least common multiple
of each variable.
Which Variable is Easier??
_________________________
More Elimination
2x + 2y = 6
3x – y = 5
Multiply the bottom equation by 2
Step 3: Multiply the equations
and solve.
Step 4: Plug back in to find the
other variable.
Step 5: Check your solution.
More Elimination
Solve the system by elimination.
x + 2y = 11
–3x + y = –5
Let’s Practice…
Solve the system using the method of your choice.
x + 2y = 11
–3x + y = –5
Let’s Practice…
Solve the system using the method of your choice.
3x + 2y = 6
–x + y = –2
Solving Systems
by Graphing
Solving a System by Graphing
Steps for Obtaining the Solution of a System of
Linear Equations by Graphing
Step 1: Graph the first equation in the system.
Step 2: Graph the second equation in the system.
Step 3: Determine the point of intersection, if any.
Step 4: Verify that the point of intersection
determined in Step 3 is a solution of the
system. Remember to check the point in
both equations.
Let’s Practice…
Solve the system by graphing. Check your answer.
y=x
y = –2x – 3
Let’s Practice…
Solve the system by graphing. Check your answer.
y = –2x – 1
y=x+5
Let’s Practice…
Solve the system by graphing. Check your answer.
𝟏
x
𝟑
y= -3
2x + y = 4
Systems & Word
Problems
Unpacking Word Problems
1. Identify each variable.
– Think: “What does each variable represent?”
2. Set up two equations.
3. Determine your method of solution.
– Think: “How can I solve the system in the easiest
way possible?”
4. Solve the system.
5. Check your answer!
Hints!!!!
Let’s Practice…
Jenna is deciding between two cell-phone plans. The first plan
has a $50 sign-up fee and costs $20 per month. The second
plan has a $30 sign-up fee and costs $25 per month. After how
many months will the total costs be the same? What will the
costs be? If Jenna has to sign a one-year contract, which plan
will be cheaper? Explain.
Let’s Practice…
One cable television provider has a $60 setup fee and $80 per
month, and the second has a $160 equipment fee and $70 per
month. In how many months will the cost be the same? What
will that cost be?
Let’s Practice…
The population of Sunny Isles is 50,000 but is growing at 2500
people per year. North Miami has a population of 26,000 but
is growing at 4000 people per year. When will both towns
have equal population?
Let’s Practice…
With a tailwind, an airplane makes a 900-mile trip in 2.25
hours. On the return trip, the plane flies against the wind and
makes the trip in 3 hours. What is the plane’s speed? What is
the wind’s speed?
Rate

Time
=
Distance
Let’s Practice…
A chemist mixes a 20% saline solution and a 40% saline
solution to get 60 milliliters of a 25% saline solution. How
many milliliters of each saline solution should the chemist use
in the mixture? Let t be the milliliters of 20% saline solution
and f be the milliliters of 40% saline solution.
20%
Saline
+
40%
Saline
=
25%
Saline
Let’s Practice…
Suppose a pharmacist wants to get 30 g of an ointment that is
10% zinc oxide by mixing an ointment that is 9% zinc oxide with
an ointment that is 15% zinc oxide. How many grams of each
ointment should the pharmacist mix together?
Solving
Systems of
Inequalities
by Graphing
Graphing Linear Inequalities
y > −𝒙 + 𝟑
Graphing Linear Inequalities
y −𝟐𝒙 > 𝟒
Details
Solid or Dotted??
•
≤ 𝒐𝒓 ≥ =
•
> 𝒐𝒓 < =
Where to Shade…
•
𝒚 > 𝒐𝒓 ≥ =
•
𝒚 < 𝒐𝒓 ≤ =
Let’s Practice…
Write an equation to represent the following:
Solving a System of Inequalities
Step 1: Re-arrange equations so y is alone, if needed.
Step 2: Graph each equation
a. Dotted or Solid?
b. Shading?
Step 3: Determine the point of intersection, if any.
Step 4: Verify that the point of intersection
determined in Step 3 is a solution of the
system. Remember to check the point in
both equations.
Let’s Practice…
Solve the system by graphing. Check your answer.
y > −𝟐𝒙 + 𝟐
y < 2x – 3
Let’s Practice…
Solve the system by graphing. Check your answer.
y ≥ −𝒙 + 𝟓
-3x + y ≤ −𝟒
Let’s Practice…
Solve the system by graphing. Check your answer.
y > 𝟐𝒙 + 𝟒
2x – y ≤ 𝟒
Let’s Practice…
Solve the system by graphing. Check your answer.
y > –2x + 5
y ≤ –2x – 4
Let’s Practice…
In one week, Ed can mow at most 9 times and
rake at most 7 times. He charges $20 for mowing
and $10 for raking. He needs to make more than
$125 in one week. Show and describe all the
possible combinations of mowing and raking that
Ed can do to meet his goal. List two possible
combinations.
Let’s Practice…
Write the system of
inequalities represented
by the graph at the right.