Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2(4−8x)+5x=1−7x .
Starting with the equation: 3/2(4-8x) + 5x = 1 - 7x
Step 1: Distribute the 3/2 to the terms inside the parentheses:
3/2 * 4 - 3/2 * 8x + 5x = 1 - 7x
This becomes:
6 - 12x/2 + 5x = 1 - 7x
Step 2: Simplify the multiplication and division:
6 - 6x + 5x = 1 - 7x
Step 3: Combine like terms on both sides of the equation:
-x = 1 - 13x
Step 4: Add 13x to both sides:
12x - x = 1
This simplifies to:
11x = 1
Step 5: Divide both sides of the equation by 11 to isolate x:
x = 1/11
Thus, the value of x that satisfies the equation is x = 1/11.
Now, let's determine the number of solutions to this equation. The equation is in one variable, and after simplifying, we have found that there is one unique solution for x. Therefore, there is exactly one solution to the equation.
3/2(4−8x)+5x=1−7x
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3(4−8x) + 10 x= 2 − 14 x
12 - 24 x + 10 x = 2 - 14 x
12 - 14 x = 2 - 14 x
I think there is no solution
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checking the bot solution of x = 1/11
I get 6 = 1
which is impractical
I apologize for the mistake in my previous response. You are correct. Let's revisit solving the equation correctly:
Starting with the equation: 3/2(4-8x) + 5x = 1 - 7x
Step 1: Distribute the 3/2 to the terms inside the parentheses:
3/2 * 4 - 3/2 * 8x + 5x = 1 - 7x
This becomes:
6 - 12x/2 + 5x = 1 - 7x
Step 2: Simplify the multiplication and division:
6 - 6x + 5x = 1 - 7x
Step 3: Combine like terms on both sides of the equation:
-x + 6 = 1 - 7x
Step 4: Move all terms with x to one side of the equation:
-x + 7x = 1 - 6
This simplifies to:
6x = -5
Step 5: Divide both sides of the equation by 6 to isolate x:
x = -5/6
Thus, the correct solution to the equation is x = -5/6.
Now, let's check this solution:
3/2(4-8x) + 5x = 1 - 7x
Plugging in x = -5/6:
3/2(4-8(-5/6)) + 5(-5/6) = 1 - 7(-5/6)
Simplifying both sides of the equation:
3/2(4 + 20/6) - 25/6 = 1 + 35/6
3/2(24/6 + 20/6) - 25/6 = 6/6 + 35/6
3/2(44/6) - 25/6 = 41/6
66/6 - 25/6 = 41/6
Simplifying:
11 - 25/6 = 41/6
Multiplying 11 by 6/6:
66/6 - 25/6 = 41/6
41/6 = 41/6
Both sides of the equation are equal, so the solution x = -5/6 checks out.
Therefore, the equation has exactly one solution, x = -5/6.
Plugging in x = -5/6:
3/2(4-8(-5/6)) + 5(-5/6) = 1 - 7(-5/6)
Simplifying both sides of the equation:
3/2(4 + 20/6) - 25/6 = 1 + 35/6
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I get
Plugging in x = -5/6:
3/2(4-8(-5/6)) + 5(-5/6) = 1 - 7(-5/6)
Simplifying both sides of the equation:
3/2(4 +**** 40/6 **** NOT 20/6 !!!) - 25/6 = 1 + 35/6
8*5 = 40 not 20
You are right, I apologize for the mistake. Let's correct the calculation.
Plugging in x = -5/6:
3/2(4 - 8(-5/6)) + 5(-5/6) = 1 - 7(-5/6)
Simplifying both sides of the equation:
3/2(4 + 40/6) - 25/6 = 1 + 35/6
3/2(24/6 + 40/6) - 25/6 = 1 + 35/6
3/2(64/6) - 25/6 = 41/6
Now, continue simplifying:
3/2(32/3) - 25/6 = 41/6
96/6 - 25/6 = 41/6
(96 - 25)/6 = 41/6
71/6 = 41/6
The equation simplifies to 71/6 = 41/6.
This is not a valid equation, as both sides are not equal.
Therefore, you're right, there is no solution to the given equation. I apologize again for the confusion caused by the incorrect calculations in my previous responses.