Solve the following systems of linear equations using any algebraic method.If possible,check your solution.
1)2x+5y=3
X-3y=7
2) 3y=11-2x
3x=y-11
3)8x-6y=14
12x-9y=18
#2:
2x + 3y = 11
3x - y = -11
multiply #2 by 3 and add:
2x+3y = 11
9x - 3y = -33
---------------
11x = -22
x = -2
now figure y: 5
Do the others likewise. To check your answers, just plug in the values and make sure the equations still balance.
To solve a system of linear equations, we can use various algebraic methods such as substitution, elimination, or graphing. I will explain how to solve each of the given systems using the substitution method.
1) 2x + 5y = 3 ...(1)
x - 3y = 7 ...(2)
To solve this system using the substitution method, we first solve one of the equations for one variable and substitute it into the other equation.
From equation (2), we can solve for x:
x = 7 + 3y ...(3)
Now, substitute equation (3) into equation (1):
2(7 + 3y) + 5y = 3
Simplify the expression:
14 + 6y + 5y = 3
11y + 14 = 3
Subtract 14 from both sides:
11y = 3 - 14
11y = -11
Divide both sides by 11:
y = -1
Now substitute y = -1 back into equation (3) to find x:
x = 7 + 3(-1)
x = 7 - 3
x = 4
Checking the solution:
Substitute x = 4 and y = -1 into the original equations:
Equation (1): 2(4) + 5(-1) = 3
8 - 5 = 3
3 = 3 (True)
Equation (2): (4) - 3(-1) = 7
4 + 3 = 7
7 = 7 (True)
Therefore, the solution to this system is x = 4 and y = -1.
2) 3y = 11 - 2x ...(4)
3x = y - 11 ...(5)
Similarly, we solve equation (5) for y:
y = 3x + 11 ...(6)
Substitute equation (6) into equation (4):
3(3x + 11) = 11 - 2x
Simplify the expression:
9x + 33 = 11 - 2x
Combine like terms:
9x + 2x = 11 - 33
11x = -22
Divide both sides by 11:
x = -2
Substitute x = -2 back into equation (6) to find y:
y = 3(-2) + 11
y = -6 + 11
y = 5
Checking the solution:
Substitute x = -2 and y = 5 into the original equations:
Equation (4): 3(5) = 11 - 2(-2)
15 = 11 + 4
15 = 15 (True)
Equation (5): 3(-2) = 5 - 11
-6 = -6 (True)
Hence, the solution to this system is x = -2 and y = 5.
3) 8x - 6y = 14 ...(7)
12x - 9y = 18 ...(8)
We can observe that equation (7) is a multiple of equation (8) by a factor of 2. This means that these two equations represent the same line and are dependent equations. As a result, the system has infinitely many solutions.
To verify this, we can divide equation (8) by 3:
(12x - 9y)/3 = 18/3
4x - 3y = 6
Now we can see that equation (7) is equivalent to equation (8) divided by 3. Therefore, these equations represent the same line, and any point that satisfies one equation will also satisfy the other equation.
To summarize, the solution to this system is infinitely many solutions.