Questions on Algebra: Linear Equations, Graphs, Slope answered by real tutors!

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# 1



Start with the given system of equations:



Multiply the both sides of the first equation by 4.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the x terms cancel out.


Simplify.


Since is NEVER true, this means that there are no solutions. So the system is inconsistent.

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# 2


Note: I've made the first equation and the second equation


Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Add to both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Combine like terms on the left side


Add 135 to both sides


Combine like terms on the right side


Divide both sides by -13 to isolate x



Reduce





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the ordered pair

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# 3




Start with the given system of equations:



Multiply the both sides of the first equation by 3.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the x terms cancel out.


Simplify.


Since is ALWAYS true, this means that there are an infinite number of solutions. So the system is consistent and dependent.

You can put this solution on YOUR website!
Hi, Hope I can help,
.
Solve by substitution or elimination method:
.
3x - 2y = 8
-12x + 8y = 32
.
Solve by substitution or elimination method:
.
7x - 5y = 14
-4x + y = 27
.
Solve by substitution or elimination method:
.
-4x + 3y = 5
12x - 9y = -15
.
First we will solve the first system with substitution
.
3x - 2y = 8
-12x + 8y = 32
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First we need to solve for a variable in one of the two equations, doesn't matter which letter, or equation, we will solve for "y" in the first equation
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, we will move (-2y) to the right side
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= = , now we need to move "8" to the left side
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= = , to find "y" we need to divide each side by "2"
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= =
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, since "y" is equal to , we can replace "y" in the other equation with , then just solve for "x"
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= , now just solve for "x"
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= = , now we will use the distribution method
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=
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Remember the + and - signs, , adding the "x"'s
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= = ( False )
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This means ( when the x's or y's cancel out, and there is a false statement) that there are no solutions, these lines are parallel, there is no intersection, and therefore no solutions
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Here is the graph of this system
.

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Now we will solve the second system, by elimination
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7x - 5y = 14
-4x + y = 27
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Elimination is when you add the two equations together, and it gets rid of a variable, first we need to make sure the x's or y's in each equation are the same, or the negative of the other
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We can eliminate any variable ( either x or y )
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We will get rid of the y's
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We need to either get both y's to "5y" or "-5y" or we need to get them to "y" or "-y", we will change the second equation to "5y"
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, to get "y" to change to "5y" we need to multiply each side by (5)
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= =
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We will need to use the distribution method
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= =
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Remember the signs, , this is our new equation
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Now we will bring the firt equation to our second new equation
.

.

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We will now add the equations
.

.

.
=
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= =
.
=
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It will become = , to find "x" we need to divide each side by
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= = = , we can now replace "x" with , in one of the two original equations
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7x - 5y = 14
-4x + y = 27
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We will use the second equation
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= = =
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Now we need to move to the right side ( we will convert "27" into "13ths"
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= = = =
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, we can check our answers by replacing "x" and "y" in both original equations
.

.

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First equation, = = = = = = ( True )
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Second equation, = = = = = , ( True )
.

.

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Solution sets are in the form (x,y), our solution set is ( , )
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The graph of the system is
.

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The intersection is your answer
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Now we can do the last system
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-4x + 3y = 5
12x - 9y = -15
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We will use the elimination method of solving this problem
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Remember elimination is where you get rid of a variable
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We will multiply the first equation by "3" to get rid of the "y"
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= = we will use distribution
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= =
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Remember signs, , now we can add the new first equation to the second equation
.

.

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= =
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= = =
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= =
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It will become = (True)
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This means ( when both x and y cancel out, and there is a true statement) that this is the same line, both equation will be one line, there is an infinite number of solutions ( since the equations are equal to one line )( or the second line is put on the first line )
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Here is the graph of this system
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Your answers are
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First system of equations = " no solutions " ( since they are parallel lines )
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Second system of equations = ( , ) ( the lines intersect at point ( , )
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Third system of equations = " infinite solutions " ( since the two equations are the same line )
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Hope I helped, Levi