Find an equation for the line with the given properties. Perpendicular to the line -2x+3y=-1;y-intercept=4

To find the equation of a line that is **perpendicular** to the given line \( -2x + 3y = -1 \) and has a **y-intercept** of 4, follow these steps:<br /><br />---<br /><br />### Step 1: Find the slope of the given line<br />The general form of a line is \( Ax + By = C \). To determine the slope of the given line, rewrite it in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope.<br /><br />Start with:<br />\[<br />-2x + 3y = -1<br />\]<br /><br />Solve for \( y \):<br />\[<br />3y = 2x - 1<br />\]<br />\[<br />y = \frac{2}{3}x - \frac{1}{3}<br />\]<br /><br />Thus, the slope of the given line is:<br />\[<br />m_1 = \frac{2}{3}<br />\]<br /><br />---<br /><br />### Step 2: Find the slope of the perpendicular line<br />The slopes of two perpendicular lines are negative reciprocals of each other. If the slope of one line is \( m_1 \), then the slope of the perpendicular line is:<br />\[<br />m_2 = -\frac{1}{m_1}<br />\]<br /><br />Substitute \( m_1 = \frac{2}{3} \):<br />\[<br />m_2 = -\frac{1}{\frac{2}{3}} = -\frac{3}{2}<br />\]<br /><br />So, the slope of the perpendicular line is:<br />\[<br />m_2 = -\frac{3}{2}<br />\]<br /><br />---<br /><br />### Step 3: Write the equation of the perpendicular line<br />We are told that the y-intercept of the desired line is 4. The slope-intercept form of a line is:<br />\[<br />y = mx + b<br />\]<br />where \( m \) is the slope and \( b \) is the y-intercept.<br /><br />Substitute \( m = -\frac{3}{2} \) and \( b = 4 \):<br />\[<br />y = -\frac{3}{2}x + 4<br />\]<br /><br />---<br /><br />### Final Answer:<br />The equation of the line is:<br />\[<br />y = -\frac{3}{2}x + 4<br />\]