Weighted Equations: Ascertain whether the equations are equivalent

To determine if the two equations are balanced, we need to evaluate them separately:

First, we solve the equation $9 - x = 1$.

• Subtract 9 from both sides to isolate $-x$:
• $-x = 1 - 9$
• $-x = -8$
• Multiply both sides by $-1$ to solve for $x$:
• $x = 8$

Next, we solve the equation $5x = -40$.

• Divide both sides by 5 to isolate $x$:
• $x = \frac{-40}{5}$
• $x = -8$

The solution to the first equation is $x = 8$ and the solution to the second equation is $x = -8$. Since the solutions for $x$ are not the same, the two equations are not balanced.

Therefore, the correct answer is No.

We will solve each equation step-by-step to ascertain if there is a common solution indicative of them being balanced:

Solving the first equation $x - 25 = 5$:

• Add 25 to both sides to isolate $x$:

$x - 25 + 25 = 5 + 25$ $x = 30$

Solving the second equation $7x - 10 = 200$:

• Add 10 to both sides to isolate the term with $x$:

$7x - 10 + 10 = 200 + 10$ $7x = 210$

• Divide by 7 to solve for $x$:

$x = \frac{210}{7}$ $x = 30$

Both equations give the solution $x = 30$. This indicates that they are indeed balanced, as they share a common solution for $x$.

Therefore, the equations are balanced.

The correct answer to the problem is: Yes.

To determine whether the given equations are balanced, we begin by examining them step-by-step:

• Given: $26 = \frac{x}{6} \stackrel{?}{=} \frac{2}{x} = \frac{1}{78}$

We should first determine the value of $x$ from the equation involving $\frac{1}{78}$:

• The last equation is $\frac{1}{78} = \frac{2}{x}$.
• Multiplying both sides by $x$ and then by $78$ gives $x = 156$.

Use $x = 156$ to evaluate the other expressions:

• For $\frac{x}{6}$:
• $\frac{156}{6} = 26$

Now, check if all values equate:

• $26 = 26$ from $\frac{x}{6}$
• $26 = 26$ from the right side of the original problem statement

All values match, indicating:

The equations are balanced. Therefore, the correct answer is Yes.