Which equations can pair with y 3x 2 to create a consistent and independent system

• Shakiyla Huggins

  Shakiyla Huggins earned an MS in Applied Mathematics from NC A&T State University and an MA in Teaching from Lenoir Rhyne University. She has teacher licensure for high school math as well as a graduate certificate in Online Teaching and Instructional Design. She has taught k-12 and college.

  Table of Contents Show

  • Consistent and Independent Systems
  • Consistent Independent System Graph
  • Independent System of Equations Examples
  • System of Equations Example 1:
  • Additional Practice - Independent Systems of Equations
  • Practice Problems
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• Instructor Mia Primas

  Mia has taught math and science and has a Master's Degree in Secondary Teaching.

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• Expert Contributor Kathryn Boddie

  Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.

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Learn what an independent system of equations is and what a consistent and independent system looks like on a graph. Find examples of independent systems of equations. Updated: 12/14/2021

A system of equations is when two or more equations are solved at the same time to determine a solution that fits both equations. The equations can be of any form (linear, quadratic, cubic, etc.), but the solution must fit both equations. In a system of equations, there are two main ways of determining the solution; graphically or algebraically. When a solution is determined graphically, the two equations must intersect at one or more points. If the two equations do not intersect at one or more points, then the systems of equations are said to have no solution. When a solution is determined algebraically, the x-value and y-value that are true for the first equation must ALSO be true for the second equation and all other equations.

Take a look at the image below to see an example of a system of equations. In this system of equations, one equation is linear {eq}y=x+3 {/eq} and the other equation is quadratic {eq}y=x^2+5 x+3 {/eq}.

Example of a System of Equations

Notice how the two equations intersect each other at two points: {eq}(-4,-1) {/eq} and {eq}(0,3) {/eq}? This means that those two coordinates are the solutions to the linear systems of equations with the equations {eq}y=x+3 {/eq} and the other equation is quadratic {eq}y=x^2+5 x+3 {/eq}. .

An independent system of equations is when the two equations can be solved graphically or algebraically and produce exactly one solution. If the system is solved graphically, then the graph of two equations intersect at exactly one point. If the system is solved algebraically, then there is exactly one x-value and y-value that satisfies both equations.

Consistent and Independent Systems

A Consistent system of Equations is when to equations have at least one solution. This means that there can be only one, or there can be more than one, but at the very least there has to be one solution. This is different from an independent system of equations, as an independent system must have exactly one solution. A system can be both consistent and independent when it has exactly one solution. However, it cannot be both consistent and independent if there is more than one solution.

Consistent Independent System Graph

The graph below is an image of a system of equations that is both consistent and independent.

Consistent and Independent System Example

The graph has two different equations: {eq}y=2x-7 {/eq} and {eq}3x=5 {/eq}. Additionally, these two different equations intersect at exactly one location, {eq}(-12,-31) {/eq}. Since there is only exactly one solution, then there is enough information to determine that this is a consistent independent system of equations.

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Independent System of Equations Examples

In the following examples, we will look at two real-world word problems which we will solve both algebraically and graphically.

System of Equations Example 1:

A large pizza at Palanzio's Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Guido's Pizza is $7.30 plus $0.65 for each topping. How many toppings need to be added to a large cheese pizza from Palanzio's Pizzeria and Guido's Pizza in order for the pizzas to cost the same, not including tax?

In this word problem, we can create two equations. The first equation is for Palanzio's Pizzeria, where the cost is $6.80 plus $0.90 for each topping. The equation that corresponds to this scenario is {eq}y=.90x+6.80 {/eq}. The second equation is for Guido's Pizza, where the cost is $7.30 plus $0.65 for each topping. The equation that corresponds to this scenario is {eq}y=.65x+7.30 {/eq}.

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Additional Practice - Independent Systems of Equations

In the following examples, students will determine whether a system is independent or dependent by examining graphs or equations and will find solutions to independent systems.

Practice Problems

1. Determine if the system of equations is independent. If it is independent, and the lines are not parallel, solve the system.

2. Determine if the system of equations is independent. If it is independent, and the lines are not parallel, solve the system.

3. Determine if the system of equations is independent. If it is independent, and the lines are not parallel, solve the system.

4. Determine if the system of equations is independent. If it is independent, and the lines are not parallel, solve the system.

Solutions

1. We will convert the second equation into slope-intercept form. Solving for y on the second equation, we have

The slope of the first line is 3 and the slope of the second line is -1, so the equations are for different lines. This system is independent and the lines are not parallel. Using substitution, we can replace y in the second equation with 3x + 1 to get

Then, substituting x = -1 into the first equation, we have

The solution is ( -1, -2 ).

2. Converting the first equation to slope-intercept form,

Converting the second equation into slope-intercept form,

The equations have a different y-intercept, so they represent different lines. Thus the system is independent. However, the lines are parallel since both have a slope of 2/3 , so there is no solution to the system.

3. The system is independent because the graph shows that the two lines are different. The solution of the system is where the lines intersect, so the solution is (4, 0).

4. Converting the second equation to slope-intercept form, we have

Both equations represent the same line, so this is a dependent system of equations.

How do you find the system of equations?

You can find a system of equations in two ways. One, by graphing, and two, algebraically. For the graphical method, find the point of intersection amongst the equations of the system. For the algebraic method, find the x-value and y-value that satisfy all equations within the system.

What is a real-life example of systems of equations?

A customer has an option to choose between paying an entrance fee each time they go to the gym or paying a one-time monthly fee. A system of equations can be used to determine how many times that customer would need to go to the gym in order to make the monthly fee worth paying

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