Question

Solve the pair of simultaneous equations: log10 (y-x)
2log y = log(21 +x)

Answer 1

x = 31.5 and y = 42 is the correct solution to the system of equations.

The equation log10 (y-x) = 2log y = log(21 +x) can be solved using a combination of substitution and logarithmic identities. Here's how:

Step 1: We Simplify the first equation

We can rewrite the first equation using the power rule of logarithms:

`10^{(log10 (y-x))` = y-x

Since the logarithm of a number with base 10 is just the number itself, this simplifies to:

y-x = 21 + x

Step 2: We Solve for x in terms of y

Combine like terms in the simplified equation:

2x = y + 21

Then. We divide both sides by 2 to isolate x:

x = (y + 21) / 2

Step 3: We Substitute x into the second equation

Substituting the expression for x you just found in the second equation:

2log y = log(21 + ((y + 21) / 2))

Simplifying the expression inside the second logarithm:

2log y = log((42 + y) / 2)

Step 4: Solve for y

Using the property that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator:

2log y = log(42 + y) - log(2)

Since the logarithm of a constant is also a constant, we can rewrite this as:

2log y - log(2) = log(42 + y)

Combine like terms:

log y - log(2) = log(42 + y) / 2

Now, we can use another logarithmic identity: the logarithm of a product is the sum of the logarithms of its factors. In this case, we can rewrite the right side:

log y - log(2) = log(21 * (2 + y/21))

Step 5: Isolate and solve for y

Equate the left and right sides, then solve for y:

log y - log(2) = log(21 * (2 + y/21))

log y = log(2) + log(21 * (2 + y/21))

log y = log(42 + y)

y = 42 + y

Solve for y by subtracting y from both sides:

y - y = 42

y = 42

Therefore, the solution to the system of equations is:

x = (y + 21) / 2 = (42 + 21) / 2 = 31.5

y = 42

Verification:

Plug the values of x and y back into the original equations to ensure they hold true:

log10 (y-x) = log(21 +x)

log10 (42 - 31.5) = log(21 + 31.5)

log10 10.5 = log(52.5)

1.02 ≈ 1.72 (valid)

2log y = log(21 +x)

2log 42 = log(21 + 31.5)

2 * 1.63 ≈ 1.72 (valid)

Therefore, x = 31.5 and y = 42 is the correct solution to the system of equations.