Download Alg 1 Sem 2 EOC Unit 2

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EOC Unit 2 Practice
Solving Systems of Equations by Graphing
Graphing is useful for solving a system of equations. Graph both equations and look for a point of intersection,
which is the solution of that system. If there is no point of intersection, there is no solution.
Example
What is the solution to the system? Solve by graphing. Check.
x+y=4
2x – y = 2
y = –x + 4
Put both equations into y-intercept form, y = mx + b
y = 2x – 2
y = –x + 4
The first equation has a y-intercept of (0, 4).
0 = –x + 4
Find a second point by substituting in 0 for y and solve for x.
x=4
You have a second point (4, 0), which is the x-intercept.
y = 2x – 2
The second equation has a y-intercept of (0, –2).
0 = 2(x) – 2
Find a second point by substituting in 0 for y and solve for x.
2 = 2x, x = 1
You have a second point for the second line, (1, 0).
2 = 2x, x = 1
You have a second point for the second line, (1, 0).
Plot both sets of points and draw both lines. The lines appear to
intersect (2,2), so (2,2) is the solution
Check
If you substitute in the point (2, 2), for x and y in your original
equations, you can double-check your answer.
x+y=4
2+2
2x – y = 2
2(2) – 2
4,
4=4
2, 2 = 2
Solving Systems of Equations by using Substitution
You can solve a system of equations by substituting an equivalent expression for one variable.
Example:
The sum of a number and twice another number is 4. The difference between twice the first number and the
second is 3. Find the two numbers.
Write and solve a system of equations: let “x” be the first number and “y” be the
second number.
Remember the word “twice” means to “double”, “sum” means addition, and
“difference” means subtraction
Then translate the sentences into symbols
x + 2y = 4
2x – y = 3
x + 2y = 4
The first equation is easiest to solve in terms of one variable.
x = 4 – 2y
Get x to one side by subtracting 2y.
2(4 – 2y) –y = 3
Substitute 4 – 2y for x in the second equation.
8 – 4y – y = 3
Distribute.
8 – 5y = 3
Simplify.
8 – 8 – 5y =3 –8
Subtract 8 from both sides
–5y = –5
Divide both sides by -5.
y=1
You have the solution for y. Solve for x.
x +2(1) = 4
Substitute in 1 for y in the first equation.
x+2–2=4–2
Subtract 2 from both sides.
x=2
The solution is (2, 1).
Substitute your solution into either of the given linear equations.
x + 2y = 4
2 + 2(1)
4=4
4
Substitute (2, 1) into the first equation.
You check the second equation
Solving Systems of Equations by using Elimination
Elimination is one way to solve a system of equations. Think about what the word “eliminate” means. You can
eliminate either variable, whichever is easiest.
Example:
A hotel is offering two weekend specials described below:
Plan 1: 3 nights and 4 meals for $233
Plan 2: 3 nights and 3 meals for $226.50
For accounting purposes, the hotel will record income for the stay and the meals separately. So the cost per
night for each special must be the same, and the cost per meal for each special mus be the same.
Write the system of equations and solve. Find the cost per night and the cost per meal.
Let “x” be the cost per night and “y” be the cost for a meal
Then 3x + 4y = 233 and 3x + 3y = 226.5
3x + 4y = 233
3x + 3y = 226.5
0x + 1y = 6.5
Subtract both equations
1y = 6.5
y = 6.5
Each meal costs $6.5
To find the cost for the night substitute 6.5 in any of the two original equations
3x + 4.(6.5) = 233
3x + 26 = 233
3x = 207
x= 69
Each night costs $69
Practice Problems:(4 points each)
1. The sum of two numbers is 19, and their difference is 55. What are the two numbers?
2. Your school committee is planning an after-school trip for 193 people to a competition at another school.
There are 8 drivers available and two types of vehicles, school buses and minivans. The school buses
seat 51 people each, and the minivans seat 8 people each. How many buses and minivans will be
needed?
3. Your local cable company offers two plans: basic service with 1 movie channel for $35 per month or basic
service with 2 movie channels for $45 per month. What is the charge for basic service and the charge for
each movie channel?
4. Two groups of students orders burritos and tacos at a local restaurant. One ordered of 3 burritos and 4
tacos costs $11.33. The other order of 9 burritos and 5 tacos costs $23.56. Wrtite a system of equations
that describes the situation. Find the cost of a burrito and the cost for a taco.
5. A farmer grows only pumpkins and corn on her 420-acre farm. This year she wants to plant 250 more
acres of corn than pumpkins. How many acres of each crop does the farmer need to plant?