7. Solve the equation \( 3(x-7)=2(x-5) \). What is the solution to the system of equations? \( \left\{\begin{array}{r}2 x-6 y=10 \\ -5 x+15 y=30\end{array}\right. \) Write an equation of a linear function with slope -2 that passes through the point \( (4,-5) \).

To solve the equation \( 3(x-7)=2(x-5) \), start by distributing both sides: \[ 3x - 21 = 2x - 10. \] Next, subtract \( 2x \) from both sides to isolate \( x \): \[ 3x - 2x - 21 = -10 \] This simplifies to \( x - 21 = -10 \). Adding 21 gives \( x = 11 \). For the system of equations: 1. From the first equation \( 2x - 6y = 10 \), solve for \( y \): Rearranging gives \( 6y = 2x - 10 \), or \( y = \frac{1}{3}x - \frac{5}{3} \). 2. From the second equation \( -5x + 15y = 30 \), simplify to \( 15y = 5x + 30 \), or \( y = \frac{1}{3}x + 2 \). Setting these equal, \( \frac{1}{3}x - \frac{5}{3} = \frac{1}{3}x + 2 \) doesn't yield a valid intersection; this means the lines are parallel, and the system has no solution. For the equation of a linear function with a slope of -2 that passes through the point (4, -5), you can use the point-slope form: \[ y - y_1 = m(x - x_1) \]. Plugging in \( m = -2 \), \( x_1 = 4 \), and \( y_1 = -5 \): \[ y + 5 = -2(x - 4) \]. Distributing gives \( y + 5 = -2x + 8 \). Subtracting 5 results in the linear function: \[ y = -2x + 3 \]. So, summarizing our solutions: \( x = 11 \), no solution for the system, and the function is \( y = -2x + 3 \).