1. \( 6 y=4 x+13 \)

Sure, let's work through the equation you've provided: \[ 6y = 4x + 13 \] **Goal:** Solve for \( y \) in terms of \( x \). ### Step 1: Isolate \( y \) To solve for \( y \), you need to isolate it on one side of the equation. Start by dividing both sides of the equation by 6: \[ \frac{6y}{6} = \frac{4x + 13}{6} \] Simplifying this, you get: \[ y = \frac{4x + 13}{6} \] ### Step 2: Simplify the Equation You can further simplify the right-hand side by breaking it into two separate fractions: \[ y = \frac{4x}{6} + \frac{13}{6} \] Simplify the fractions: \[ y = \frac{2x}{3} + \frac{13}{6} \] ### Final Result The equation solved for \( y \) is: \[ y = \frac{2}{3}x + \frac{13}{6} \] This is the slope-intercept form of the equation, where: - **Slope (\( m \))** is \( \frac{2}{3} \) - **Y-intercept (\( b \))** is \( \frac{13}{6} \) ### Graphical Interpretation If you were to graph this equation on a coordinate plane: - The line would slope upwards to the right with a slope of \( \frac{2}{3} \). - It would intersect the y-axis at \( y = \frac{13}{6} \) (which is approximately 2.1667). ### Additional Considerations If you need to find specific values of \( y \) for given \( x \) values or vice versa, you can substitute the known value into the equation and solve for the unknown. **Example:** Find \( y \) when \( x = 3 \). \[ y = \frac{2}{3}(3) + \frac{13}{6} = 2 + \frac{13}{6} = \frac{25}{6} \approx 4.1667 \] Feel free to ask if you have more questions or need further assistance with this equation!