ratios in systems of equations

Since this is not my area of expertise, I searched Google under the key words "ratios 'systems of equations'" to get these possible sources:

http://en.wikipedia.org/wiki/Simultaneous_equation
http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Solving-Systems-of-Equations-Simultaneous-Equations-.topicArticleId-9046,articleId-9042.html
http://www.explorelearning.com/View/correlations/Textbooks/Books/BCElemandInterAlg2002.html
http://canmin.geoscienceworld.org/cgi/content/abstract/32/4/969

In the future, you can find the information you desire more quickly, if you use appropriate key words to do your own search.

I hope this helps. Thanks for asking.

When dealing with systems of equations, ratios can be useful in a variety of ways. Ratios can help us compare different quantities, find unknowns, or express relationships between variables.

To begin, let's understand what a ratio is. A ratio represents the relative sizes of two or more quantities. It is typically expressed in the form of "a:b" or as a fraction "a/b." For example, if we have a ratio of 2:3, it means that the first quantity is two-thirds of the second quantity.

In systems of equations, ratios can help us solve for unknowns by expressing the relationship between different variables in terms of their ratios.

Here's a step-by-step guide on how to use ratios in systems of equations:

1. Define the variables: Start by assigning variables to the quantities you want to compare. Let's consider a system of two equations:
Equation 1: 2x + 3y = 10
Equation 2: 4x + 6y = 20

2. Choose a variable: Select one variable (let's say x) and express the other variable (y) in terms of the chosen variable. This will allow us to determine the ratio between x and y.
Let's solve Equation 1 for y: 2x + 3y = 10
Rearranging the equation, we get: 3y = 10 - 2x
Divide both sides of the equation by 3: y = (10 - 2x)/3

3. Express the ratio: Now, we have an expression for y in terms of x. We can express the ratio between x and y by dividing the coefficients of x and y in this expression.
The ratio of x to y is given by: x/y = 1/((10 - 2x)/3)

4. Substitute the ratio: Substitute the ratio obtained in step 3 into the other equation of the system. This will eliminate one variable and allow us to solve for the remaining variable.
Substitute x/y = 1/((10 - 2x)/3) into Equation 2: 4x + 6y = 20
Replace x/y with 1/((10 - 2x)/3): 4x + 6*(1/((10 - 2x)/3)) = 20

5. Solve for the remaining variable: Simplify the equation obtained in step 4 and solve for the remaining variable (in this case, y).

The process described above can be used to solve systems of equations using ratios. However, keep in mind that this is just one method, and there are various other techniques for solving systems of equations, such as substitution, elimination, and graphing. The choice of method depends on the specific problem and your preferred approach.