Line \( t_{1} \) has the equation \( y=x+5 \) and line \( t_{2} \) has the equation \( y=x+2 \). Find the distance between \( t_{1} \) and \( t_{2} \). Round your answer to the nearest tenth.

To find the distance between the two parallel lines \( t_{1} \) and \( t_{2} \), we can use the formula for the distance between two parallel lines given by the equations \( y = mx + b_1 \) and \( y = mx + b_2 \), which is: \[ \text{Distance} = \frac{|b_2 - b_1|}{\sqrt{1 + m^2}} \] In this case, for line \( t_{1} \), \( b_1 = 5 \) and for line \( t_{2} \), \( b_2 = 2\). The slope, \( m = 1 \). Now, substituting the values into the distance formula: \[ \text{Distance} = \frac{|2 - 5|}{\sqrt{1 + 1^2}} = \frac{|-3|}{\sqrt{2}} = \frac{3}{\sqrt{2}} \approx 2.1213 \] Rounding this to the nearest tenth, the distance between the two lines is approximately \( 2.1 \).