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

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3 Answers

 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 )
 
 
 
 
 
 
  
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:
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.