Finding Point of Intersection of Two Lines
To find the coordinates of the point of intersection of the two lines, we first need the equations of both lines.
The equation of the first line is given as \( 2x - 3y + 1 = 0 \).
The second line passes through the points S(7,-6) and T(-3,2). We can find the slope of this line (m) using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
\[ m = \frac{2 - (-6)}{-3 - 7} \]
\[ m = \frac{8}{-10} \]
\[ m = -\frac{4}{5} \]
Next, we will use the point-slope form to write the equation for the second line using point S(7, -6) or T(-3, 2). I'll choose S(7, -6) for this example:
\[ y - y_1 = m(x - x_1) \]
\[ y - (-6) = -\frac{4}{5}(x - 7) \]
\[ y + 6 = -\frac{4}{5}x + \frac{4}{5} \cdot 7 \]
\[ y + 6 = -\frac{4}{5}x + \frac{28}{5} \]
\[ y = -\frac{4}{5}x + \frac{28}{5} - \frac{30}{5} \]
\[ y = -\frac{4}{5}x - \frac{2}{5} \]
To find the intersection, we solve the two equations together. Let's write them in the standard form:
Equation of line 1: \( 2x - 3y + 1 = 0 \)
Equation of line 2: \( \frac{-4}{5}x - y - \frac{2}{5} = 0 \)
Now multiply equation 2 by 5 to remove fractions:
Equation of line 2: \( -4x - 5y - 2 = 0 \)
To solve the system, we can use either substitution or elimination. Let's use elimination:
Multiply the first equation by 4 to align coefficients for x:
\[ 8x - 12y + 4 = 0 \]
\[ -4x - 5y - 2 = 0 \]
Add these equations together:
\[ 4x - 17y + 2 = 0 \]
\[ 2 \rightarrow 8x - 17y = -2 \]
Now, we will solve for \( x \) using equation 1 and substitution into equation 2:
\[ 2x - 3y + 1 = 0 \rightarrow 2x = 3y - 1 \]
\[ x = \frac{3}{2}y - \frac{1}{2} \]
Substitute \( x \) in equation \( 8x - 17y = -2 \):
\[ 8(\frac{3}{2}y - \frac{1}{2}) - 17y = -2 \]
\[ 12y - 4 - 17y = -2 \]
\[ -5y = 2 \]
\[ y = \frac{-2}{5} \]
Now substitute \( y \) back into one of the original equations to find \( x \):
\[ 2x - 3\left(\frac{-2}{5}\right) + 1 = 0 \]
\[ 2x + \frac{6}{5} + 1 = 0 \]
\[ 2x + \frac{6}{5} + \frac{5}{5} = 0 \]
\[ 2x = -\frac{6}{5} - \frac{5}{5} \]
\[ 2x = -\frac{11}{5} \]
\[ x = -\frac{11}{10} \]
Therefore, the coordinates of the point of intersection of the two lines are \(x = -\frac{11}{10}\) and \(y = \frac{-2}{5}\), or more clearly written as \((-1.1, -0.4)\).
