An equation is shown below:
3 over 2 multiplied by x plus 7 over 2 equals 2 to the power of x
What is the solution to the equation
A.X=1
B.X=3
C.X=5
D.X=8
I think it could be d
Let X = 3:
3x/2 + 7/2 = 2^x.
3*3/2 + 7/2 = 2^3.
9/2 + 7/2 = 8.
16/2 = 8.
8 = 8.
So X = 3 satisfies the Eq.
To solve the given equation step-by-step, we can use algebraic manipulation:
1. The equation is: (3/2) * x + (7/2) = 2^x.
2. Multiply both sides of the equation by 2 to eliminate the fractions:
2 * [(3/2) * x + (7/2)] = 2 * 2^x
3x + 7 = 2^(x+1)
3. Rewrite the right side of the equation using exponential properties:
3x + 7 = 2 * 2^x
4. Rewrite 2 as 2^1:
3x + 7 = 2 * 2^x
3x + 7 = 2^1 * 2^x
3x + 7 = 2^(x+1)
5. Since the bases on both sides of the equation are equal, we can equate the exponents:
x + 1 = x + 1
6. Subtract x from both sides of the equation:
1 = 0
7. Since 1 is not equal to 0, this equation has no solution.
Therefore, none of the options A, B, C, or D are correct. The solution to the equation is that there is no solution.
To solve this equation, we need to isolate the variable "x" on one side of the equation. Let's go through the steps one by one:
1. Start with the given equation: 3/2 * x + 7/2 = 2^x
2. Subtract 7/2 from both sides of the equation to isolate the term containing "x". After this step, the equation becomes: 3/2 * x = 2^x - 7/2
3. To simplify the equation further, we need to express the right side of the equation with the same base as the left side. In this case, let's rewrite 2^x as (2^(x/2))^2. So, the equation now becomes: 3/2 * x = (2^(x/2))^2 - 7/2
4. Now, we can square both sides of the equation to eliminate the exponent on the right side: (3/2 * x)^2 = ((2^(x/2))^2 - 7/2)^2
Simplifying further: (9/4 * x^2) = ((2^(x/2))^2 - 7/2)^2
5. At this stage, we can substitute the values from the options (A, B, C, D) and see which one satisfies the equation.
Let's start with option D, X = 8:
- Substitute X = 8 into the equation: (9/4 * 8^2) = ((2^(8/2))^2 - 7/2)^2
- Simplify the equation further and see if both sides are equal.
- If they are equal, then option D, X = 8, is the solution to the equation.
Note: It can be quite time-consuming to manually evaluate each option and substitute it back into the equation. Alternatively, you can use a graphing calculator or numerical methods to find the solution.