Question 590617: These 3 problems are causing me stress as I can barely understand the elimination and substitution methods for working these.
For the systems of linear equations
Determine how many solutions exist
Use either elimination or substitution to find the solutions (if any)
Graph the two lines, labeling the x-intercepts, y-intercepts, and points of intersection
y = 2x + 3 and y = -x - 4
2x + 3y = 8 and 3x + 2y = 7
I have more questions like these, but I think if I could please receive some help I could get a better idea of how to work the rest of them. Even if no one answers this thanks for reading, sincerely me
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your first 2 equations are in slope intercept form already.
they are:
y = 2x + 3 and y = -x - 4
since they are both equal to y, you can set them equal to each other to get:
2x + 3 = -x - 4
add x to both sides of the equation and subtract 3 from both sides of the equation to get:
3x = -7
divide both sides of the equation by 3 to get:
x = -7/3
substitute -7/3 for x in either equation to solve for y.
use y = 2x + 3 which becomes:
y = -14/3 + 3 which becomes:
y = -14/3 + 9/3 which becomes:
y = -5/3
substitute for x and y in the other equation of y = -x - 4 to get:
-5/3 = -(-7/3) - 4 which becomes:
-5/3 = 7/3 - 12/3 which becomes:
-5/3 = -5/3 confirming the values for x and y are good.
your second set of equations is in standard form which can be solved using substitution or elimination.
the equations are:
2x + 3y = 8 and 3x + 2y = 7
you could solve for y in both equations and use the method described above but that is not necessary.
solving by substitution:
your equations are:
2x + 3y = 8
3x + 2y = 7
solve for x or y in one of the equations and then use that value to substitute in the other equation as shown below:
take the first equation and solve for y:
2x + 3y = 8
subtract 2x from both sides of the equation to get:
3y = 8 - 2x
divide both sides of the equation by 3 to get:
y = (8-2x)/3
substitute that value for y in the second equation.
3x + 2y = 7 becomes:
3x + 2*(8-2x)/3 = 7
simplify to get:
3x + 16/3 - 4x/3 = 7
multiply both sides of the equation by 3 to get:
9x + 16 - 4x = 21
combine like terms to get:
5x + 16 = 21
subtract 16 from both sides of the equation to get:
5x = 5
divide both sides of the equation by 5 to get:
x = 1
use that value for x in the first equation to get:
2x + 3y = 8 becomes:
2 + 3y = 8
subtract 2 from both sides of the equation to get:
3y = 8-2 which becomes:
3y = 6
divide both sides of the equation by 3 to get:
y = 2
substitute 1 for x and 2 for y in the second equation to get:
3x + 2y = 7 becomes:
2 + 4 = 7 which becomes:
7 = 7
this confirms the values of 1 for x and 2 for y are good.
these values apply to both equations at the same time (simultaneously).
they are a common solution to both equations.
2x + 3y = 8 becomes 2 + 6 = 8 which becomes 8 = 8.
3x + 2y = 7 becomes 3 + 4 = 7 which becomes 7 = 7.
x = 1 and y = 2 is a common solution to both equations.
solving by elimination:
your equations are:
2x + 3y = 8
3x + 2y = 7
you are going to add these equations together or you are going to subtract one of these equations from the other to eliminate one of the variables which will allow you to solve for the remaining variable.
the process that allows you to do this is the ability to multiply or divide both sides of each equation without changing the equality that is already there as represented by the equal sign.
in this system of equations, i want to eliminate the x variable.
in order to do that i need to multiply the first equation by 3 and the second equation by 2.
after doing, that my equations become:
6x + 9y = 24 (both sides of first equation multiplied by 3)
6x + 4y = 14 (both sides of second equation multiplied by 2)
now i will subtract the second equation from the first to get:
5y = 10
i will now divide both sides of this equation by 5 to get:
y = 2
i will now use the value of 2 for y in either equation to solve for x.
taking the first original equation (use the original equation for this just in case you messed up when you multiplied the original equation to get the modified equation).
the original equations are:
2x + 3y = 8 (first equation)
3x + 2y = 7 (second equation)
substituting 2 for y in the first equation gets:
2x + 3y = 8 becomes:
2x + 6 = 8
subtract 6 from both sides of the equation to get:
2x = 2
divide both sides of the equation by 2 to get:
x = 1
substitute 2 for y and 1 for x in the second equation to get:
3x + 2y = 7 becomes:
3 + 4 = 7 which becomes:
7 = 7
this confirms the values for x and y are good.
your solution is x = 1 and y = 2
this solutions applies to both equations simultaneously.
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