if ab=3.bs, cd=2x+3,and ba=x+1. If ad=31 units, find x, ab, bc and cd
To find the values of x, ab, bc, and cd, we can use the given equations and the properties of equality. Let's break down each step to solve this problem:
1. Start with the equation ab = 3.bs. Here we are given that ab is equal to 3 times bs. We need to find the value of ab, so we need to isolate it on one side of the equation.
- Divide both sides of the equation by 3: ab/3 = bs.
- Thus, we have ab/3 = bs.
2. Use the equation ba = x + 1. This equation tells us that ba is equal to x + 1.
- Since ab and ba are the same, we can substitute ab for ba.
- So, ab = x + 1.
3. Now, we have two equations: ab/3 = bs and ab = x + 1. We can combine these equations to solve for x and find the values of ab, bc, and cd.
- Substitute ab from equation 2 into equation 1: (x + 1)/3 = bs.
- Multiply both sides of the equation by 3 to eliminate the fraction: x + 1 = 3bs.
- Rearrange the equation: 3bs = x + 1.
4. Next, we can use the equation cd = 2x + 3. This equation tells us that cd is equal to 2x + 3.
- Since cd and ad are the same, we can substitute cd for ad.
- So, cd = 2x + 3.
5. We are also given that ad = 31 units. Substitute cd for ad in the equation: 2x + 3 = 31.
- Solve for x: 2x = 31 - 3.
- Simplify: 2x = 28.
- Divide both sides by 2: x = 14.
6. Now plug the value of x into the equations to find the values of ab, bc, and cd.
- ab = x + 1 = 14 + 1 = 15.
- bc = ab = 15.
- cd = 2x + 3 = 2(14) + 3 = 31.
Therefore, the values are:
- x = 14
- ab = 15
- bc = 15
- cd = 31