# Systems of Equations Project

### By: Shiva Jayaraman

## What is a system of equations and what does the solution represent?

## Method of Graphing- 500 points

## Substitution- 500 points

## Elimination- 500 points

## Word Problem- 500 points

*A rental company charges a flat fee of x dollars for a floor sander rental plus y dollars per hour of the rental. One customer rents a floor sander for 4 hours and pays $63. Another customer rents a floor sander for 6 hours and pays $87. Find the flat fee and the cost per hour for the rental.*

For this problem, I used the method of elimination.

## Solving- Part 1 Here, I found the Y-Coordinate. | ## Solving- Part 2 Here, I found the X-Coordinate. | ## Solving- Part 3 Here, I found the ordered pair solution. |

## Write a System- 500 points

*You pay $24.50 for 10 gallons of gasoline and 1 quart of oil at a gas station. Your friend pays $22 for 8 gallons of the same amount of gasoline and 2 quarts of the same oil. Find the cost of 1 quart of oil.*

## When it's best to use Graphing

Example:

y=x+3

y=2x+4

Here, both equations are in slope-intercept form, so to graph them, we would find the slope and y-intercept of both equations, plug those in to make the graph, and find the intersection point of the two lines. Then, we find the solution by using the point of intersection.

## When it's best to use Substitution

Example:

2x+y=5

y=x-4

Here, the second equation shows what y is equal to, and in the top equation, the coefficient is 1 for y, therefore making it just y. You can substitute the bottom equation into the y in the top equation. Both equations are solved for the same variable, or either equation is solved for a variable.

## When it's best to use Elimination

Example:

3x + 3y =15

2x - 3y = 5

Here, you can subtract the y values, and they cancel out, since the top is 3 and the bottom is -3. 3-3=0, so they cancel out. Now, you can find the x values, and solve for the y values once that part is complete. All you would need to do once that happens, is to substitute the x value that you found from elimination, and plug it into the equation to figure out y.

## 3 Different Types of Solutions

**No Solution:**If the system of equations have the same slope, but different y-intercepts, then we will not have an intersection point, so there won't be a solution. They'll be parallel lines on a graph.

**Infinitely Many Solutions:** If the system of equations has the same slope and same y-intercept, they will be on the exact same line, so there won't be just one particular intersection point.

**One Solution:** The two equations will have different slopes, therefore having only one intersection point on the graph.

## No Solution As you can see, in the graph, the lines are parallel, so there isn't any intersection. Therefore, it has no solution. | ## Infinitely Many Solutions Here, it looks like only one line is there, but it's really two lines that have the same slope and same y-intercept. This way, they are on the exact same line, and there are infinitely many solutions for the coordinates. | ## One Solution In this graph, the lines have different slopes, so there's only one intersection point. Only one solution will be there to the equation. |