4. Plant Growth Analysis (Systems of Linear Equations): - Two types of plants will be grown in the garden: Type A and Type B. The growth rate of Type A is given by the equation \( y=2 x+5 \) and for Type B by \( y=-x+15 \), where \( x \) is the number of weeks since planting, and \( y \) is the height in centimeters. - Determine after how many weeks the heights of Type A and Type B plants will be the same. Solve the system of linear equations to find this intersection point.

Solve the system of equations \( y=2x+5;y=-x+15 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y=2x+5\\y=-x+15\end{array}\right.\) - step1: Substitute the value of \(y:\) \(2x+5=-x+15\) - step2: Move the variable to the left side: \(2x+5+x=15\) - step3: Add the terms: \(3x+5=15\) - step4: Move the constant to the right side: \(3x=15-5\) - step5: Subtract the numbers: \(3x=10\) - step6: Divide both sides: \(\frac{3x}{3}=\frac{10}{3}\) - step7: Divide the numbers: \(x=\frac{10}{3}\) - step8: Substitute the value of \(x:\) \(y=2\times \frac{10}{3}+5\) - step9: Simplify: \(y=\frac{35}{3}\) - step10: Calculate: \(\left\{ \begin{array}{l}x=\frac{10}{3}\\y=\frac{35}{3}\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=\frac{10}{3}\\y=\frac{35}{3}\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(\frac{10}{3},\frac{35}{3}\right)\) The system of linear equations represents the growth rates of Type A and Type B plants. The solution to the system is \( (x,y) = \left(\frac{10}{3},\frac{35}{3}\right) \) or in decimal form \( (x,y) = (3.\dot{3},11.\dot{6}) \). This means that after approximately 3.33 weeks, the heights of Type A and Type B plants will be the same.