Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Ivan and his workout partner are lifting weights together, doing many sets of each exercise. On a certain exercise, Ivan is using an 11-pound bar, increasing the amount of weight he lifts by 12 pounds on each set. His partner, meanwhile, started out using a 10-pound bar and is upping the weight by adding 13 pounds on every set. Eventually, Ivan and his workout partner will be lifting the same amount, and will take turns using the same barbell. How many sets will they have completed? How much weight will they be lifting then? After completing \( \square \) sets, they will both be lifting \( \square \) pounds.

To describe the situation with a system of equations, we can define the following variables: - Let \( x \) be the number of sets completed. - Let \( I \) be the weight Ivan is lifting after \( x \) sets. - Let \( P \) be the weight his partner is lifting after \( x \) sets. From the problem, we know: - Ivan starts with an 11-pound bar and increases his weight by 12 pounds for each set. Therefore, the equation for Ivan's weight is: \[ I = 11 + 12x \] - His partner starts with a 10-pound bar and increases his weight by 13 pounds for each set. Therefore, the equation for his partner's weight is: \[ P = 10 + 13x \] We want to find the number of sets \( x \) when both Ivan and his partner are lifting the same amount of weight, which gives us the equation: \[ 11 + 12x = 10 + 13x \] Now, we can solve this equation for \( x \): 1. Subtract \( 12x \) from both sides: \[ 11 = 10 + x \] 2. Subtract 10 from both sides: \[ 1 = x \] So, \( x = 1 \). This means they will have completed 1 set. Next, we can find out how much weight they will be lifting after 1 set by substituting \( x = 1 \) back into either equation. We'll use Ivan's equation: \[ I = 11 + 12(1) = 11 + 12 = 23 \] Now, let's check his partner's weight to ensure they match: \[ P = 10 + 13(1) = 10 + 13 = 23 \] Both Ivan and his partner will be lifting 23 pounds after 1 set. Thus, we can fill in the blanks: After completing \( \text{1} \) sets, they will both be lifting \( \text{23} \) pounds.