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linear equation

Solve the systems of linear equations by using the method of substitution. If the system is consistent, inconsistent, or dependent, say so in your answer. y=1/3x +10         x-3y=30

Comments

Matthew, use the parentheses when you write the fractions. What you wrote is
         1                         
y  =   —  +  10 , and this is not a linear function.   
         3x  

better will be y = (1/3)x + 10 , or y = 1/3 x + 10

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4 Answers

y = (1/3)x + 10

x - 3y = 30   Substitute  to get x - 3[(1/3)x +10] = 30

                   Simplify:  x - (x + 30) = 30

                                 x - x - 30 = 30

                                        -30 = 30  no solution. Inconsistent

                                

Comments

First find the easiest operation between the 2:

y= (1/3)x +10 (1)

x - 3y = 30      (2)

(2) is the easiest because you can have x = 3y +30 <=> x = 3(y+10)

Now replace X with his value on (1)

y = (1/3)[3(y+10)] <=> y = y+10 <=> y-y=10. There is no Solutions

The system is inconsistent.

Let's first understand what the question is asking us to find. We need to find the solutions (solution set) using substitution method. This means that there are several ways to answer this question, but we must demonstrate our knowledge of substitution to answer this question and it is probably your best bet. Second, after we find our solution set, we must figure out if it is consistent or inconsistent. Meaning, there are two ways to determine the solution set. Consistent means that we have at least one solution that will fit upon our the linear functions or that they intersect, in this case y = (1/3)x + 10 and x - 3y = 30. Inconsistent means that there weren't any solutions that fits upon our linear functions or no intersections. Now, this will be said for independency or dependency of the graph. If the functions are the same line on the graph, then it is considered dependent. Otherwise, it is independent if there are more than one line on the graph. Now that we have a better understanding of what we are looking for, lets find our solutions.

 

y = (1/3)x + 10

x - 3y = 30

1. Let's go ahead and use the first equation, y = (1/3)x + 10 and substitute this into our y value of the second equation or linear function. The equal sign also means that our y is a third of x plus 10. 

x -3[(1/3)x + 10) = 30

2. We will then multiply or distribute the 3 among the (1/3)x + 10.

x - x - 30 = 30

Note: 1/3 is the reciprocal of 3. This means that 3 * 1/3 = 1. It's easier to remember that 3 * 1/3 = 3/3 = 1. 

3. We see that x - x. Well, a positive + its negative = 0. In other words: x + (-x) = 0. We are now left with:

-30 = 30

4. Well, had we needed to find the absolute value of both sides, then yes, they would be equal because the absolute value is the distance from zero on the number line; however, this is actually stating that negative 30 is positive 30. Obviously, that is inaccurate and therefore, the system is inconsistent. The graph will also show that the functions never intersect. However, there are two different linear functions when graphed, and therefore they are independent systems.

y = (1/3)x + 10 ....... (1)
x - 3y = 30 ---> y = (1/3)x - 10 ..... (2)
There is the same slope but are different coordinates of y-intercept, thus graphs of two equations are parallel this means there are no solutions. In this case the system is inconsistent.

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