use all three methods graphing,substitution,linear combination to solve 3x+4y=5 and -2x+y=4

Question

after state how many solutions there are and give a reason 

please explain 

thanks

Answer 1

 Elimination Method:

 

- 2X + y = 4

  3X  + y =4

 

    3X + 4Y = 5

-4(-2X- 4Y  = 4 ) 

   11 X = 5 -16 = -11

        X = -11/ 11 = -1

 

 Substitute X = -1 into 2nd equation

 

- 2( -1) + Y = 4

 

    Y = 4 -2 =2

 

 Substitution method

      3X + 4Y = 5

      -2X + Y = 4

    

     Y = 2X +4     from 2nd equation:

 

      3X + 4( 2X + 4) = 5

   

      11X + 16 = 5

 

       11X = -11     X = -1

 

      Substitute in 2nd equation:

 

        -2( -1) + Y = 4

 

          Y = 2 

 

 

  To graph :

 

   3X + 4Y = 5        3( 0) + 4Y = 5      Y intercept = 5/4     ( 0 , 5/4)

 

                             3 X + 4 (0 ) = 5    X intercept = 3/5    ( 3/5, 0)    

        

                               Connect the 2 points together, and have the graph

      

     - 2X + Y = 4         -2( 0) + Y = 4      ( 0 , 4 )  is Y intercept.

                                 -2X  + 0  = 4       (-2, 0 )  is X intercept

                               Connect 2 points and get the graph of the line

             Observe that intersect point is :

 

                                              ( -1 ,2 )

 

 

 

 

 

 

  

Answer 2

3x+4y=5 => y = (5 - 3x)/4 = -3/4 x + 5/4

-2x+y=4 => y = (4 + 2x)/1 = 2 x + 4

 

The slopes are different, so the graphs will be two lines that intersect in one point, called the "solution" of the system of equations.

 

Solving by substitution:

 

y = -3/4 x + 5/4 = 2 x + 4

 

Multiply by 4:

 

-3 x + 5 = 8 x + 16

 

Add 3x-16 to both sides:

 

-11 = 11x

 

x = -1

 

y = 2 (-1) + 4 = 2

 

So the solution is (-1,2).

 

Solving by elimination:

 

3x+4y=5 => 3x+4y=5
-2x+y=4 => 8x-4y=-16

                   11x = -11

x = -1

 

3x+4y=5 => 6x+8y=10
-2x+y=4 => -6x+3y=12
                        11y = 22

y = 2

 

So the solution, (-1,2), is the same as for substitution.

 

Solving using Cramer's Rule:

D = | 3  4 | = 3 - -8 = 11

      | -2 1 |

 

D_x = | 5 4 | = 5-16 = -11
          | 4 1 |

 

D_y = |  3 5 | = 12 - -10 = 22
          | -2 4 |

 

x = D_x / D = -11/11 = -1

 

y = D_y / D = 22/11 = 2

 

So solution, as before, is (-1,2).

 

Solve by graphing:

 

One way is to convert each equation into Intercept Form:

 

ax + by = c => x/(c/a) + y/(c/b) = 1

 

where the intercepts are below their variable. Then graph the two intercepts and draw the line through them.

 

3x+4y=5 => x/(5/3)+y/(5/4)=1
-2x+y=4 => x/(4/-2)+y/(4/1)=1

 

See GeoGebra sketch here:

http://www.wyzant.com/resources/files/260091/intersecting_lines_for_student_answer

Answer 3

Hi Angelica;
3x+4y=5 and -2x+y=4
LINEAR COMBINATIONS
Both equations are linear combinations in that these are in the format of Ax+By, A and B are constants multiplying variables x and y.
3x+4y=5 is in Standard Formula...
Ax+By=C, neither A nor B equal zero and A is greater than zero.
-2x+y=4 is NOT in Standard Formula. A is less than zero. Let's fix that by multiplying both sides by -1...
(-1)(-2x+y)=(4)(-1)
2x-y=-4

SUBSTITUTION
3x+4y=5 and 2x-y=-4
Let's take either equation and isolate a variable. Obviously, it would be easiest to take the second equation and isolate y...
2x-y=-4
Let's subtract 2x from both sides...
2x-2x-y=-2x-4
-y=-2x-4
Let's multiply both sides by -1...
(-1)(-y)=(-1)(-2x-4)
y=2x+4
Let's take the first equation and substitute y with 2x+4...
3x+4y=5
3x+[(4)(2x+4)]=5
3x+8x+16=5
11x+16=5
Let's subtract 16 from both sides...
11x+16-16=5-16
11x=-11
Let's divide both sides by 11...
(11x)/11=-11/11
x=-1
Let's plug this into either equation to establish the value of y. I select the original second equation. It is easiest...
-2x+y=4

[(-2)(-1)]+y=4

2+y=4

y=2

Let's take both x and y results and plug these into the first equation for verification...

3x+4y=5

[(3)(-1)]+[(4)(2)]=5

-3+8=5

5=5

 

GRAPHING

I cannot do such here.

However,

3x+4y=5

2x-y=-4

The slope of each equation is -A/B...

3x+4y=5, -(3/4)=-3/4.

2x-y=-4, -(2/-1)=2

The y-intercept can be easily established as x=0...

3x+4y=5, 4y=5, y=5/4, y-intercept, (0,5/4)

2x-y=-4, -y=-4, y=4, y-intercept, (0,4)

When graphing, begin with the y-intercept.  This is the point at which the line crosses the y-axis.  For the first line, the line will increase 3 units as it runs to the left 4 units.  For the second line, the line will increase 2 units as it runs to the right 1 unit.  The two lines will insect at (-1,2).

 

 

ELIMINATION

This is another method you do not mention.

3x+4y=5 and -2x+y=4

To do this, either variable must have the same coefficient.  Currently, x has the coefficients of 3 and -2, whereas y has the coefficient of 4 and 1.

Let's take the second equation.

-2x+y=4

Let's multiply both sides by 4.

On second thought, let's multiply both sides by -4 such that we convert this into Standard Formula...

(-4)(-2x+y)=(4)(-4)

8x-4y=-16

Let's add the two equations together and eliminate...

8x-4y=-16

+(3x+4y=5)

11x=-11

 

x=-1

 

 

 

SUBSTITUTION, GRAPHING AND ELIMINATION ARE ALL TECHNIQUES WHICH CAN BE USED TO SOLVE THIS.