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# Systems of equations

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I have a few questions I need help with.

What are the steps to solving a system of equation by elimination?

What are the steps to solving a system of equation by substitution?

What are the steps to solving a system of equation by graphing?

Once you have graph a system of equation, how do you check your answer?

When we solve a system of linear equation we get a solution.  It is important to understand what the solution tells us about the system.  Keep that in mind, in your own words what is a inconsistent system?

When we solve a system of linear equation we get a solution.  It is important to understand what the solution tells us about the system.  Keep that in mind, in your own words what is an undetermined system?

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## SOLUTION This solution is FREE courtesy of BrainMass!

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Please find attached the solution properly described in the attached file.

Question -

What are the steps to solving a system of equation by elimination?

There are usually Five steps to solve a system of equation by elimination:-

1. Firstly modify both the given equations in the standard form, Ax + By = C.
2. Determine which variable to eliminate with Addition or Subtraction.
3. Solve for the variable that is left after doing the addition or subtraction operation.
4. Go back and use the found variable in step 3 to find second variable.
5. Check the solution in both equations of the system.

Ex-:
Solve the following system using addition.

2x + y = 9
3x - y = 16

Note that, if I add down, the y's will cancel out. So I'll draw an "equals" bar under the system, and add down:

2x + y = 9
3x - y = 16
5x = 25

Now I can divide through to solve for x = 5, and then back-solve, using either of the original equations, to find the value of y. The first equation has smaller numbers, so I'll back-solve in that one:

2(5) + y = 9
10 + y = 9
y = -1

Then the solution is (x, y) = (5, -1)

Question -

What are the steps to solving a system of equation by substitution?

For Substitution we need to follow these steps:-

1. Choose any variable x or y.
2. Then try to convert one of the equations in such a way that you get value of one variable in terms of another variable. Like y = Ax + B...or something of this sort.
3. This gives equation only in one variable form. Solve for that variable.
4. Put the value of variable in either of original equation and get another variable.
5. Check the result by placing both values in original equation.

Consider the example:-

Solve this system of equations and check:

3y - 2x = 11
y + 2x = 9

1. Solve one of the equations for either "x =" or "y =".

This example solves the second equation for "y =".

3y - 2x = 11
y = 9 - 2x

2. Replace the "y" value in the first equation by what "y" now equals. Grab the "y" value and plug it into the other equation.

3(9 - 2x) - 2x = 11

3. Solve this new equation for "x".

(27 - 6x) - 2x = 11
27 - 6x - 2x = 11
27 - 8x = 11
-8x = -16
x = 2

Place this new "x" value into either of the ORIGINAL equations in order to solve for "y". Pick the easier one to work with!

y = 9 - 2(2)
y = 9 - 4
y = 5

5. Check: substitute x = 2 and y = 5 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE!

3y - 2x = 11
3(5) - 2(2) = 11
15 - 4 = 11
11 = 11 (check!)

y + 2x = 9
5 + 2(2) = 9
5 + 4 = 9
9 = 9 (check!)

Question -

What are the steps to solving a system of equation by graphing?

Steps of solving by graphics include:-

1. Draw a line for the given equations by taking various points (x, y)
randomly that satisfy the given equations.
2. The lines of equations if intersect, that intersection point is called solution.
3. Note the point (x, y). It is the required solution.

Consider the following example to have better understanding -

Solve the system of equations by graphing. 2x+y=1. and x+2y=-2.
Steps for answer -
Solve each equation for y
2x+y=1 ===> y=-2x+1

x+2y=-2 ===> y=(-x/2) -2

then set a table for x and y values, choose any value for x to find y
such as (for first equation)
if x=1, then y=2*1+1=3, your ordered pair is (1,3)
if x=0, then y=2*0+1 = 1, (0,1) ....etc
then put them on graph and draw a line,
the same thing for second equation, draw a line and see if they intersect, and at which point.

Hope you know how to plot ordered pair on a graph

Question -

Once you have graph a system of equation, how do you check your answer?

When you have plotted the lines to get the required solution note the point (x, y) from the graph at which the two lines intersect (or meet). That is the solution (of x and y).
Put it on any original equation to check and it will always satisfy.

You can get 3 possible types of graphs -
1. In which the 2 lines are intersecting each other, in such cases the solution is said to be one solution. It has only intersection point and the system of such equations is called as independent system.
2. In second case, you may have pair of parallel lines, such system has no solution and they are called inconsistent system. They have no intersection point.
3. The third case will be such, in which, both the lines will be overlapping each other, so such type of system is called dependent system. And the solution of such system is the whole line.

Question -

When we solve a system of linear equation we get a solution. It is important to understand what the solution tells us about the system. Keep that in mind, in your own words what is a inconsistent system?

An inconsistent system of equation refers to that system of equation that doesn't have
any solution. We can't solve these type of system by above given methods.

For example:-

Below given set of equations don't have any solution.

x-y=2

4x-4y=5

And, if you try to solve such system of equations graphically you will get a pair of lines which are parallel to each other. Since parallel lines never cross, then there can be no intersection; that is, for a system of equations that graphs as parallel lines, there can be no solution. This is called an "inconsistent" system of equations, and it has no solution.

Question -

When we solve a system of linear equation we get a solution. It is important to understand what the solution tells us about the system. Keep that in mind, in your own words what is an undetermined system?

An undetermined system of equation is such a system where we cannot determine the values of the variables because we are not given enough set of equations to solve.

It generally contains more variables than the number of equations.

For Ex:-

3x+4y+z=6
x+y=5

The above set of equations are undetermined as we require one more equation to find value of z.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

© BrainMass Inc. brainmass.com October 4, 2022, 1:19 pm ad1c9bdddf>
https://brainmass.com/math/linear-algebra/algebra-solving-graphing-systems-equations-251927