Ever wonder how to make word problems easier to solve? Here are some keywords:

Addition

• Sum
• Total
• Altogether
• In all
• Combined
• Increase
• Plus
• Added to
• More than

Example: “There are 12 apples and 8 oranges. How many fruits are there in all?”

Subtraction

• Difference
• Left
• Fewer than
• Less than
• Subtracted from
• Decrease
• Take away
• Minus
• How many more/less?

Example: “Sara has 15 candies, and she gave 7 to her friend. How many does she have left?”

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Multiplication

• Product
• Times
• Multiplied by
• Of (e.g., “half of”)
• Twice, thrice
• Each (when referring to groups)
• In total (when repeated addition is implied)

Example: “A box has 6 rows of 5 chocolates. How many chocolates are there in total?”

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Division

• Quotient
• Divided by
• Equal parts
• Per
• Out of
• Average
• Ratio
• Split/shared equally

Example: “There are 24 candies, and 6 friends want to share them equally. How many candies does each friend get?”

Steps for the Substitution Method

1. Choose an Equation and Solve for One Variable:
   • From one of the equations in the system, isolate one variable (ex: x,y….)
2. Substitute the Expression:
   • Replace the isolated variable in the other equation(s) with the expression obtained in Step 1.
3. Simplify and Solve:
   • Solve the resulting equation, which now has only one variable.
4. Back-Substitute to Find the Other Variable:
   • Use the value obtained in Step 3 and substitute it back into the expression from Step 1 to find the value of the other variable.
5. Check Your Solution:
   • Substitute the values of both variables into the original equations to verify the solution.

 

Try to solve: x+y=7    2x-y=4     using substitution

 

 

General Form: 3x−y−6+5=0 —- 3x−y−1=0

Y intercept form: y=3x−1

From this project, I have learned absolutely nothing. This project was very challenging as it required many steps as well as inputting many numbers into DESMOS. The graphing took many tries as the first 2 hours of trying to input the numbers weren’t working and the YouTube videos weren’t helping either, after trying random formulas, it worked. The easiest part of this project was when I input the numbers into DESMOS and took a screen shot of my finished work.

An undefined slope occurs when a line is perfectly vertical. In such cases, the “run” (horizontal change) is zero, which makes the slope impossible to calculate mathematically.

Vertical lines:

• For a vertical line, the x-coordinate of every point on the line is the same.
• This means Δx=0 because there’s no horizontal change.
• When you plug 0 into the denominator of the slope formula, you get:

1. Rise:
   • The vertical change between two points on a line.
   • It shows how much the line goes up or down.
   • A positive rise means the line goes upward, and a negative rise means the line goes downward.
2. Run:
   • The horizontal change between two points on a line.
   • It shows how far the line moves left or right.
   • Run is always measured as a positive distance.

How to tell if something is a function or a relation:

A relation is any set of ordered pairs. These pairs consist of inputs and outputs, (x, y). Relations describe how one set (the domain) is associated with another set (the range).

Example of a Relation:

{(1,2),(2,3),(3,4),(2,5)} See how the #2 is paired with #3 and #5. This is an example of how the #2 is a relation as it has more than one partner.

A function is a special type of relation with a specific rule: each input is related to exactly one output: no input value is paired with more than one output value.

Example of a Function:

{(1,2),(2,3),(3,4)} Each input 1, 2 and 3 is paired with exactly one output.

 

The domain of a function is the set of all possible input values (or “x” values). It’s the collection of values you can plug into the function without causing any issues, like division by zero or taking the square root of a negative number.

The range is the set of all possible output values (or “y” values) based on the inputs from the domain. It represents all the values that the function can output when you use every value in the domain.

– The domain is all real numbers because you can square any real number.
– The range is all non-negative real numbers (y ≥ 0) because squaring any real number cannot produce a negative result.

Desmos is a website that can be used to show the ways that a number sequence and or pattern can be graphed to show how it looks on a graph. Desmos shows the points on a graph where the numbers entered touches on the Y and X axis. This website is a great way for people (students and teachers) to be able to check their work as it gives a visual representation of how their work is shown on a graph.

Here is an example of an equation put into desmos:

This shows how the points on the answer show on the Y and X axis.