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

after state how many solutions there are and give a reason
thanks

### 3 Answers by Expert Tutors

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
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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 )

Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
5.0 5.0 (3 lesson ratings) (3)
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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

-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:
Vivian L. | Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACHMicrosoft Word/Excel/Outlook, essay comp...
3.0 3.0 (1 lesson ratings) (1)
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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.