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.