Question

Given the equations:

2q - r = 4p
2r = q
r = 2p
Prove that 1 = r.
Given the equations:

3c + d = f + g
d = f
g = c + 8
Prove that c = 4.
a) Both equalities are true.
b) Only the first equality is true.
c) Only the second equality is true.
d) None of the equalities are true.

Answer 1

To prove that 1 = r, substitute the given values for q, r, and p into the equations and simplify. To prove that c = 4, substitute the given values for d, f, and g into the equation and simplify.

Step-by-step explanation:

To prove that 1 = r, we need to substitute the given values for q, r, and p into the equations and simplify. Given that q = 2r and r = 2p, we can substitute these values into the equation 2q - r = 4p. Substitute 2r for q: 2(2r) - r = 4p. Simplify: 4r - r = 4p. Combine like terms: 3r = 4p. Since we also know that r = 2p, we can substitute this value into the equation: 3(2p) = 4p. Simplify: 6p = 4p. Subtract 4p from both sides: 6p - 4p = 4p - 4p. Simplify: 2p = 0. Divide both sides by 2: p = 0. Substitute this value back into the equation r = 2p: r = 2(0). Simplify: r = 0. Therefore, 1 = r.

To prove that c = 4, we follow the same steps. Given that d = f, g = c + 8, and 3c + d = f + g, we can substitute these values into the equation. Substitute d for f: 3c + d = d + g. Substitute g for c + 8: 3c + d = d + (c + 8). Simplify: 3c + d = d + c + 8. Combine like terms: 3c = c + 8. Subtract c from both sides: 3c - c = c + 8 - c. Simplify: 2c = 8. Divide both sides by 2: c = 4. Therefore, c = 4.