Systems of Equations Project

By: Shiva Jayaraman

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

A system of equations is a set of two or more equations containing two or more variables. The solution represents an ordered pair that satisfies all the equations in the system. You can either eliminate, substitute, or even graph to figure out your solution. Your solution will either be no solution, infinitely many solutions, or one solution.

Method of Graphing- 500 points

With the Method of Graphing in systems of equations, we graph both equations in the same coordinate system. The solution is the point where the two lines intersect. In the graph below where I worked out the solution, I found the slope and y-intercept, and made the lines so that they match the equation's slope. I also made sure its y-intercept matched.

Substitution- 500 points

The goal when using substitution is to reduce the system to one equation that has only one variable. Then, we can solve the equation to find this variable, and then substitute the value in the other equation to find the other variable.

Elimination- 500 points

Like Substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two equations in the system together. Align all like terms together and then determine whether any like terms can be eliminated because they have opposite coefficients. Eliminate one of the variables and solve for the other variable. Use the value of the variable in one of the original equations and solve for the other variable.

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.

Write a System- 500 points

Here, I had to solve a word problem by making a system of equations based off of the problem. When I figured it out, I solved the System of Equations.


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

When both equations are solved for y (the slope-intercept form), and/or you want to estimate a solution, graphing would work best.


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

Sometimes, it's difficult to find the exact solution by graphing. Look for a variable in either equation that has a coefficient of either 1 or -1. Then, substitution will work best, because we can easily decide which equation to solve for x or y.


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

When both equations have the same variable, with the same or opposite coefficients, or if the variable term in one equation is the multiple of the corresponding variable term in the other equation.


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.