justify your steps using the properties of equality and determine the number of solutions to the equation 3/2 (4-8x) + 5x = 1 - 7x
To solve the equation 3/2 (4-8x) + 5x = 1 - 7x, we can follow these steps using the properties of equality:
1. Distribute the 3/2 to the terms inside the parentheses:
(3/2)(4) - (3/2)(8x) + 5x = 1 - 7x
This simplifies to:
6 - 12x/2 + 5x = 1 - 7x
2. Simplify the terms on the left side:
6 - 6x + 5x = 1 - 7x
Combine like terms:
6 - x = 1 - 7x
3. Get rid of the variables by isolating x on one side:
Move the terms with x to the left side:
-x + 7x = 1 - 6
Combine like terms:
6x = -5
4. Solve for x by dividing both sides by 6:
x = -5/6
Therefore, the solution to the equation is x = -5/6.
Since we obtained a specific value for x, there is only one solution to the equation.
To solve the equation 3/2(4-8x) + 5x = 1 - 7x, we will use the properties of equality to simplify and determine the number of solutions.
Step 1: Distribute the 3/2 to the terms inside the parentheses on the left side of the equation.
3/2(4-8x) + 5x = 1 - 7x
6 - 12x + 5x = 1 - 7x
Step 2: Combine like terms on both sides of the equation.
-12x + 5x + 7x = 1 - 6
0 = -5
Step 3: Analyze the resulting equation with no variable terms (0 = -5).
Since the equation 0 = -5 is not true, this implies that there are no solutions to the original equation.
Therefore, the number of solutions to the equation 3/2(4-8x) + 5x = 1 - 7x is 0, indicating that it is an inconsistent equation.
To justify each step using the properties of equality, we will simplify the equation step by step as follows:
Step 1: Distribute the 3/2 to the terms within the parentheses.
(3/2) * 4 - (3/2) * 8x + 5x = 1 - 7x
Simplifying: 6 - 12x/2 + 5x = 1 -7x
Simplifying further: 6 - 6x + 5x = 1 - 7x
Step 2: Combine like terms.
6 - 6x + 5x = 1 - 7x
Simplifying: 6 - x = 1 - 7x
Step 3: Add 7x to both sides of the equation.
6 - x + 7x = 1 - 7x + 7x
Simplifying: 6 + 6x = 1
Step 4: Subtract 6 from both sides of the equation.
6 + 6x - 6 = 1 - 6
Simplifying: 6x = -5
Step 5: Divide both sides of the equation by 6.
(6x)/6 = (-5)/6
Simplifying: x = -5/6
By simplifying the equation using the properties of equality, we determined that the solution to the equation 3/2 (4-8x) + 5x = 1 - 7x is x = -5/6.
Therefore, the equation has only one solution.