Consider the linear pattern: \( x ; 29 ; y ; 41 \) Calculate the values of \( x \) and \( y \).

To find the values of \( x \) and \( y \) in the linear pattern \( x ; 29 ; y ; 41 \), we can use the concept of arithmetic sequences. Given: - The pattern is an arithmetic sequence with a common difference between consecutive terms. - The first term is \( x \). - The second term is 29. - The third term is \( y \). - The fourth term is 41. Let's denote the common difference as \( d \). The formula for an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \] where: - \( a_n \) is the nth term in the sequence, - \( a_1 \) is the first term, - \( n \) is the term number, - \( d \) is the common difference. We can use this formula to find the values of \( x \) and \( y \) by setting up equations based on the given terms in the pattern. 1. For the second term (29): \[ 29 = x + d \] 2. For the third term (\( y \)): \[ y = 29 + d \] 3. For the fourth term (41): \[ 41 = y + d \] Now, we can solve these equations to find the values of \( x \) and \( y \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}29=x+d\\y=29+d\\41=y+d\end{array}\right.\) - step1: Substitute the value of \(y:\) \(\left\{ \begin{array}{l}29=x+d\\41=29+d+d\end{array}\right.\) - step2: Simplify: \(\left\{ \begin{array}{l}29=x+d\\41=29+2d\end{array}\right.\) - step3: Solve the equation: \(\left\{ \begin{array}{l}29=x+d\\d=6\end{array}\right.\) - step4: Substitute the value of \(d:\) \(29=x+6\) - step5: Swap the sides: \(x+6=29\) - step6: Move the constant to the right side: \(x=29-6\) - step7: Subtract the numbers: \(x=23\) - step8: Substitute the value of \(d:\) \(y=29+6\) - step9: Simplify: \(y=35\) - step10: Calculate: \(\left\{ \begin{array}{l}d=6\\x=23\\y=35\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}d=6\\x=23\\y=35\end{array}\right.\) - step12: Rewrite: \(\left(d,x,y\right) = \left(6,23,35\right)\) The solution to the system of equations is: \[ d = 6, x = 23, y = 35 \] Therefore, the values of \( x \) and \( y \) in the linear pattern are \( x = 23 \) and \( y = 35 \).