Question - Finding Point of Intersection of Two Lines
Solution:
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 $$(-1.1, -0.4)$$.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com