A shoe company makes 7 pairs of white shoes for every 4 pairs of black shoes. If it makes 121,000 pairs of white shoes and black shoes combined, how many pairs are white?

Let's denote the number of pairs of white shoes as \( W \) and the number of pairs of black shoes as \( B \). Given that the company makes 7 pairs of white shoes for every 4 pairs of black shoes, we can express this ratio as: \[ \frac{W}{B} = \frac{7}{4} \] We are also given that the company makes a total of 121,000 pairs of white and black shoes combined. This can be expressed as: \[ W + B = 121,000 \] Now, we have a system of two equations: 1. \( \frac{W}{B} = \frac{7}{4} \) 2. \( W + B = 121,000 \) We can solve this system of equations to find the number of pairs of white shoes. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\frac{W}{B}=\frac{7}{4}\\W+B=121000\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}\frac{W}{B}=\frac{7}{4}\\B=121000-W\end{array}\right.\) - step2: Substitute the value of \(B:\) \(\frac{W}{121000-W}=\frac{7}{4}\) - step3: Cross multiply: \(W\times 4=\left(121000-W\right)\times 7\) - step4: Simplify the equation: \(4W=7\left(121000-W\right)\) - step5: Expand the expression: \(4W=847000-7W\) - step6: Move the variable to the left side: \(4W+7W=847000\) - step7: Add the terms: \(11W=847000\) - step8: Divide both sides: \(\frac{11W}{11}=\frac{847000}{11}\) - step9: Divide the numbers: \(W=77000\) - step10: Substitute the value of \(W:\) \(B=121000-77000\) - step11: Calculate: \(B=44000\) - step12: Calculate: \(\left\{ \begin{array}{l}B=44000\\W=77000\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}B=44000\\W=77000\end{array}\right.\) - step14: Rewrite: \(\left(B,W\right) = \left(44000,77000\right)\) The solution to the system of equations is \( B = 44,000 \) and \( W = 77,000 \). Therefore, the number of pairs of white shoes is 77,000.