Elimination method 

To solve a system of equations by elimination, start with both equations in standard form. Then decide which variable will be the easiest to eliminate. You want to have the coefficients of one variable to be opposites so that you can add the equations together and eliminate that variable.

Example 1

3x+y=5

2x-y=0

Add the equations together. The y variable cancels out.

Example 2 

x+4y=2

2x+5y=-2

Multiply the first equation by -2 to get the x variables opposite of each other.

The x variable cancels out.

Isolate the y by dividing both sides by -3 to find the value of y.

Why

Both substitution and elimination are ways to solve systems of equations. Though there are some systems where it is easier to use the elimination method rather than substitution. By knowing how to solve the system using elimination it expands your options of ways to solve the system.

Vocabulary

Standard form- Ax+By=C

Finding slope

A way to find the slope of two coordinates is to create a table chart and plug in the points under x and y.

Example 1

Find the slope of (0,3) and (1,5)

Step 1:  Put the x and y points of the two coordinates into the table

Step 2: As x increases by 1 each time, y increases by 2.

Step 3: Put these numbers into the format of rise over run.

The slope is 2.

Example 2 

Find the slope of (2,5) and (4,11)

Step 1:  Put the x and y points of the two coordinates into the table

Step 2: As x increases by 2 each time, y increases by 6.

Step 3: Put these numbers into the format of rise over run.

The slope is 3.

Why

Using a table chart to find the slope of two coordinates is an easy and efficient way to find the slope.

Vocabulary

Slope- The slope of a line is a measure of its steepness. Slope is calculated as “rise over run”.

Problem solving

Step 1: Introduce variables to represent unknown values.

Step 2: Form a system of equations using the variables.

Step 3: Solve the system.

Step 4: Answer the problem and verify the solution.

Example 1

Your are running a concession stand, and are selling hot dogs and sodas. Each hot dog costs $1.50 and each soda costs $0.50. At the end of the day you made a total of $78.50. You sold a total of 87 hot dogs and sodas combined. You must report the number of hot dogs sold and the number of sodas sold. How many hot dogs were sold and how many sodas were sold?

Step 1 

x = the number of hot dogs sold

y= the number of sodas sold

Step 2

1.50x + 0.50y = 78.50

x + y = 87

Step 3 

Solve the system.

(35, 52) is the solution to the system.

Step 4

35 hot dogs were sold and 52 sodas were sold.

Verify answer.

If both equations check properly then the answers are correct.

Example 2 

You are going to get lunch with a friend. You order three tacos and three burritos and your total bill is $11.25. Your friend’s bill is $10.00 for four tacos and two burritos. How much do tacos cost and how much do burritos cost?

Step 1 

x= price of 1 taco

y= price of 1 burrito

Step 2

3x + 3y = 11.25  (Equation representing your lunch)

4x + 2y = 10   (Equation representing your friend’s lunch)

Step 3 

Solve the system.

The solution to the system is x= 1.25 and y= 2.5.

Step 4

One taco costs $1.25 and one burrito costs $2.50.

Verify answer.

Why

By knowing how to find solutions to word problems it helps to find unknown variables that can be in real life situations. For an example, if you are trying to find the cost of something that is not clearly stated, you can easily find out by using your knowledge of systems of equations.

From completing this task I learnt that it is easier and more efficient to find the line using the point-slope form and then finding the general form and y-intercept form after. Some challenges I encountered was finding that my lines were not lining up. I solved these challenges by looking at the coordinates to make sure it is correct, and if not, I changed the coordinates so that it would work. Overall, this assignment was fun to do and I enjoyed it.

Domain- The domain of a relation is the set of all possible values that can be used for the input of the independent variable (x).

Range- The range of a relation is the set of all possible values of the output of the dependent variable (y).

How to find domain

Step 1: Identify the domain of the relation. Which is along the x axis.

Step 2: The domain of the relation starts at -3 and ends at 1.

Step 3: Write the domain as follows: -3 < x ≤ 1

Step 4: This shows all the possible values that can be used for the input.

How to find range

Step 1: Identify the range of the relation. Which is along the y axis.

Step 2: The range of the relation starts at -4 and ends at 0.

Step 3: Write the range as follows: -4 ≤ x ≤ 0.

Step 4: This shows all the possible values that can be used for the output.

Why

Range and domain is used in real life to make mathematical calculations. For an example, range can be used to calculate the amount of time that has passed and we can take a function that represents real world situations and then analyze what the domain and range represent in the function. Domain and range are important values to help define a relation.

Sources

Learning, Lumen. “College Algebra.” Lumen, https://courses.lumenlearning.com/waymakercollegealgebra/chapter/find-domain-and-range-from-a-graph/.

Function Notation 

In math, function notation is used to show the independent variable in a function for example, f(x) means that the value of the function f depends on the value of the independent variable x.

Example 1

f(x) = 2x + 3

Input 5 into the x.

f(5) = 2(5) + 3

Write the equation with 5 where the x’s are.

f(5) = 13

Solve the equation.

The symbol f(x) provides a formula for the function f, and represents the value of the function for a given value of x.

Example 2 

f(x) = 5x – 7

Input 2 into the x.

f(2) = 5(2) – 7

Write the equation with 2 where the x’s are.

f(2) = 3

Solve the equation.

Why

Function notation is an efficient way to write functions that is easy to read and understand. It allows you to easily see the input value for the independent variable inside the parentheses.

Vocabulary

Independent variable- A variable that represents a quantity that does not depend on any other variable for its value.

Function- A mathematical relationship between two variables, where every input variable has one output variable.

X-Intercept

Where a line crosses the x-axis on a graph.

Y-Intercept 

Where a line crosses the y-axis on a graph.

How to find the x-intercept

To find the x-intercept set y= 0 and solve the equation for x.

Ex. `y= 2x – 8

step 1: Set the y as 0.

step 2: Isolate the x. When we bring the x over to the opposite side of the equal sign it changes into a negative number.

step 3: Divide both sides by -2 to leave only the x variable.

step 4: Once you have solved it, the number remaining is what x is equal to.

How to find the y-intercept 

To find the y-intercept set x = 0 and solve for y.

Ex. y= 3x -15

step 1:Set the x as 0.

step 2: The Y is already isolated, so we only need to -15 from 0 and we get -15.

Why

It’s important to know how to solve equations to find the x and y intercepts because it is a huge part of graphing and having the skill to find the intercepts quickly will make graphing easier. It will also help in finding patterns.