These are all Called Systems of Equations in Mathematics:
Q1.). Solve: x – y = 1, –3x + 3y = 3 ===> There is No Solutions
I will try to solve your equations:
x - y = 1;
- 3x + 3y = 3
Step 1: Solve
x - y = 1 for x:
x - y + y = 1 + y (Add: Y, to both sides:)
x = y + 1
Step 2: Substitute Y + 1 for X in - 3x + 3y = 3
- 3x + 3y = 3
-3 ( y + 1) + 3y = 3
- 3 = 3 (Simplify both sides)
- 3 + 3 = 3 + 3 (Add 3 to both sides):
When Graph: y = x - 1; y = x + 1
0 = 6 ===> Therefore, Answer, No Solutions to this Equation.
Q2). Solve: 2y = 3 – 4x, x + y = 0
Solve: your system of equation:
x + y = 0; 2y = -4x + 3
Step 1: Solve: x + y = 0 for x:
x + y + - y = 0 + - y ( Add - y, to both sides)
x = - y
Step 2: Substitute: - y for x in 2y = - 4x + 3:
2y = - 4x + 3
2y = - 4 ( - y ) + 3
2y = 4y + 3 ( Simplify both sides of equation: ):
2y + - 4y = 4y + 3 + - 4y ( Add - 4y to both sides):
- 2y = 3
- 2y / 2 = 3 / - 2 ( Divide both sides by - 2 )
y = - 3 / 2
Step 3: Substitute: - 3 / 2 for y in x = - y
x = - y
x = - -3 / 2
x = 3/2 (Simplify both sides)
When Graph: y = - 2x + 1.5; y = -x
Answer: ====> x = 3 / 2; and y = -3 / 2
Q 3:). Solve: 12x + 2 = 4y – 10; 2x + y = –11
Solve equation:
2x + y = - 1; 12x + 2 = 4y - 10
Step 1: Solve: 2x + y = - 1 for y:
2x + y + -2x = - 1 + - 2x ( Add -2x to both sides ):
y = -2x - 1
Step 2: Substitute: - 2x - 1 for y in 12x + 2 = 4y - 10:
12x + 2 = 4y - 10
12x + 2 = 4 ( - 2x - 1 ) - 10
12x + 2 = -8x - 14 ( Simplify Both Sides:):
12x + 2 + 8x = -8x - 14 + 8x ( Add 8x to both sides:):
20x + 2 = -14
20x + 2 -2 = -14 + -2 ( Add -2 to both sides:):
2ox = -16
20x / 20 = -16 / 20 (Divide both sides by 20:):
x = -4 / 5
Step 3: Substitute: - 4 / 5 for X in y = - 2x - 1:
y = -2x - 1
y = -2 ( -4 / 5 ) - 1
y = 3 / 5 ( Simplify both sides):
When Graph: y = 3x + 3; y = -2x - 1
Therefore, Answer, y = 3 / 5 and x = -4 / 5
Q4.). Solve: –3x – y = 5; 15x = 10 – 5y ====> There is No Solutions:
Solve your equation:
-3y - y = 5; 15x = -5y + 10
Step 1: Solve: -3x - y = 5 for y:
-3x - y + 3x = 5 + 3x ( Add 3x to both sides:):
-y = 3x + 5
-y / -1 = 3x + 5 / -1 (Divide both sides by -1):
y = -3x - 5
Step 3: Substitute: -3x - 5 for y in 15x = -5y + 10:
15x = -5y + 10
15x = -5 ( -3x - 5) + 10
15x = 15x + 35 ( Simplify both sides):
15x + -15x = 15x + 35 + -15x ( Add -15x Both sides):
0 = 35
When Graph: y = -3x - 5; y = -3x + 2
Therefore, Answer, No Solutions to this Equation:
Q 5.). Solve: x + y = 1; –4x + 2y = 7
Solve your Equation:
x + y = 1; -4x + 2y = 7
Step 1: Solve: x + y + -y = 1 + -y ( Add -y both sides ):
x = -y + 1
Step 2: Substitute: -y + 1 for X in -4x + 2y = 7
-4x + 2y = 7
-4 ( - y + 1 ) + 2y = 7
6y - 4 = 7 (Simplify both sides):
6y - 4 + 4 = 7 + 4 ( Add 4 to both sides):
6y = 11
6y / 6 = 11 / 6 ( Divide both sides by 6 ):
y = 11 / 6
Step 4: Substitute: 11 / 6 for y in X = -y + 1
x = -y + 1
x = - 11 / 6 + 1
x = -5 / 6 (Simplify both sides ):
When Graph: y = -x + 1; y = 2x + 3
Therefore, Answer, x = -5 / 6 and y = 11 / 6
Hence, Answers to your Questions: (Q1: x – y = 1, –3x + 3y = 3 There is No Solutions) , and (Q4: –3x – y = 5, 15x = 10 – 5y, There is No Solutions.). Both, Q1; and Q4, has No Solutions to the Systems of Equations:
Hope that Helps!!!! : )
