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.