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 using the properties of equality, we will follow these steps:
Step 1: Distribute the 3/2 to the terms inside the parentheses:
(3/2)(4) - (3/2)(8x) + 5x = 1 - 7x
Step 2: Simplify the expression using the distributive property:
6 - 12x/2 + 5x = 1 - 7x
Step 3: Simplify the expression further by dividing 12x by 2:
6 - 6x + 5x = 1 - 7x
Step 4: Combine like terms on both sides of the equation:
-6x + 5x + 7x = 1 - 6
Step 5: Simplify the expression by adding and subtracting like terms:
6x = -5
Step 6: Divide both sides of the equation by 6 to solve for x:
x = -5/6
Therefore, the equation 3/2(4−8x)+5x=1−7x has a single solution, which is x = -5/6.
Step 1: Distribute the 3/2 to the terms inside the parentheses:
(3/2)(4) - (3/2)(8x) + 5x = 1 - 7x
Step 2: Simplify the expression by multiplying:
(6/2) - (24/2)x + 5x = 1 - 7x
Step 3: Combine like terms:
3 - 12x + 5x = 1 - 7x
Step 4: Add 7x to both sides of the equation to isolate the variables on one side:
3 - 12x + 5x + 7x = 1 - 7x + 7x
Step 5: Combine like terms:
3 - 12x + 12x = 1
Step 6: Simplify the expression:
3 = 1
Step 7: Since the equation 3 = 1 does not hold true, there is no solution to the equation 3/2(4−8x)+5x=1−7x.
To justify our steps using the properties of equality, we need to perform the same operations on both sides of the equation. Let's simplify and solve the equation step by step:
1. Distribute the 3/2 to (4-8x):
(3/2)*(4) - (3/2)*(8x) + 5x = 1 - 7x
6/2 - 12x/2 + 5x = 1 - 7x
3 - 6x + 5x = 1 - 7x
2. Combine like terms on the left side:
-x = 1 - 7x
3. Add 7x to both sides:
7x - x = 1 - 7x + 7x
6x = 1
4. Divide both sides by 6:
(6x)/6 = 1/6
x = 1/6
Now, let's determine the number of solutions to the equation based on the value of x:
Since we obtained a specific value for x, which is 1/6, the equation has ONE solution.