Line \( \ell_{1} \) has the equation \( -x+y=4 \) and line \( \ell_{2} \) has the equation \( -x+y=-3 \). Find the distance between \( \ell_{1} \) and \( \ell_{2} \). Round your answer to the nearest tenth.

To determine the distance between the two parallel lines \( \ell_{1} \) and \( \ell_{2} \), we can use the formula for the distance between two parallel lines given in the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \): \[ \text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \] First, we convert the given line equations into the appropriate form. For \( \ell_{1} \): \[ -x + y - 4 = 0 \quad \Rightarrow \quad A = -1, B = 1, C_1 = -4 \] For \( \ell_{2} \): \[ -x + y + 3 = 0 \quad \Rightarrow \quad C_2 = 3 \] Now we can plug in the values into the distance formula: \[ \text{Distance} = \frac{|3 - (-4)|}{\sqrt{(-1)^2 + 1^2}} = \frac{|3 + 4|}{\sqrt{1 + 1}} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2} \] Calculating this gives: \[ \frac{7\sqrt{2}}{2} \approx \frac{7 \times 1.414}{2} \approx \frac{9.899}{2} \approx 4.949 \] Rounding this to the nearest tenth, we find that the distance between the lines is approximately: \[ \boxed{4.9} \]