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 \( (4,-5) \).

Let's solve each of the problems step by step. --- **1. Solve the equation \( 3(x - 7) = 2(x - 5) \):** \[ \begin{align*} 3(x - 7) &= 2(x - 5) \\ 3x - 21 &= 2x - 10 \quad \text{(Expand both sides)} \\ 3x - 2x &= -10 + 21 \quad \text{(Subtract \(2x\) and add \(21\) to both sides)} \\ x &= 11 \end{align*} \] **Solution:** \( x = 11 \) --- **2. Solve the system of equations:** \[ \begin{cases} 2x - 6y = 10 \\ -5x + 15y = 30 \end{cases} \] First, simplify both equations: - **First Equation:** \[ 2x - 6y = 10 \\ \Rightarrow x - 3y = 5 \quad \text{(Divide both sides by 2)} \] - **Second Equation:** \[ -5x + 15y = 30 \\ \Rightarrow x - 3y = -6 \quad \text{(Divide both sides by -5)} \] Now, the system becomes: \[ \begin{cases} x - 3y = 5 \\ x - 3y = -6 \end{cases} \] This system is **inconsistent** because the same left-hand side equals two different right-hand sides. **Solution:** No solution (the system is inconsistent). --- **3. Write an equation of a linear function with slope -2 that passes through the point \( (4, -5) \):** We'll use the **point-slope form** of a linear equation: \[ y - y_1 = m(x - x_1) \] Where: - \( m = -2 \) (slope) - \( (x_1, y_1) = (4, -5) \) Plugging in the values: \[ y - (-5) = -2(x - 4) \\ y + 5 = -2x + 8 \\ y = -2x + 3 \] **Equation of the linear function:** \( y = -2x + 3 \) --- **Summary of Solutions:** 1. \( x = 11 \) 2. No solution (the system is inconsistent) 3. \( y = -2x + 3 \)