Straight Lines and Coordinates

The equation of a line in slope-intercept form is given by the formula y = mx + b, where m is the slope of the line and b is the y-intercept. Therefore, the equation of the line that passes through the point (3, 4) and has a slope of 2 is y = 2x + b. To find the value of b, we can substitute the point (3, 4) into the equation: 4 = 2(3) + b. Solving for b, we get b = 2. Therefore, the equation of the line is y = 2x + 2.

To find the equation of the line, we can use the point-slope form: (y - y_1 = m(x - x_1)), where ((x_1, y_1)) is one of the given points and (m) is the slope of the line. Using the point ((2, 3)), we have: (y - 3 = m(x - 2)). To find the slope, we can use the other given point ((5, 7)): (7 - 3 = m(5 - 2)). Simplifying this, we get (m = 2). Substituting this value of (m) back into the point-slope form, we get the equation of the line: (y - 3 = 2(x - 2)), which can be simplified to (y = 2x + 1).

To find the equation of the line, we can use the point-slope form: (y - y_1 = m(x - x_1)), where ((x_1, y_1)) is a point on the line and (m) is the slope of the line. Using the two given points, we can find the slope: (m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{5 - 2} = 2). Plugging in the point ((2, 3)) and the slope (m = 2), we get the equation of the line: (y - 3 = 2(x - 2) = 2x - 4). Therefore, the equation of the line is: (y = 2x - 1).

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1)/(x2 - x1). Therefore, the slope of the line passing through the points (2, 3) and (5, 7) is (7 - 3)/(5 - 2) = 2.

The equation of a line passing through a point (x1, y1) and having a slope of m is given by the formula y - y1 = m(x - x1). Therefore, the equation of the line passing through the point (3, 4) and having a slope of -2 is y - 4 = -2(x - 3), which simplifies to y = -2x + 10.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula m = (y2 - y1) / (x2 - x1). Substituting the given values, we get m = (7 - 3) / (5 - 2) = 4 / 3 = 2.

The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Substituting the given values, we get y = -3x + b. To find the value of b, we can use the point (2, 5). Substituting x = 2 and y = 5, we get 5 = -3(2) + b. Solving for b, we find b = 11. Therefore, the equation of the line is y = -3x + 11.

The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Given that the slope is -3 and the line passes through the point (2, 5), we can substitute these values into the equation to find the y-intercept: 5 = -3(2) + b, which gives b = 11. Therefore, the equation of the line is y = -3x + 11.

The angle between two lines can be found using the formula: angle = arctan(|m2 - m1| / (1 + m1 * m2)), where m1 and m2 are the slopes of the lines. Substituting the slopes of the given lines, we get: angle = arctan(|2 - (-1)| / (1 + 2 * (-1))) = arctan(3 / (-1)) = arctan(-3) ≈ 45°. Therefore, the angle between the lines is approximately 45°.

The equation of a line in three-dimensional space is given by $x = A + Bt$, $y = C + Dt$, $z = E + Ft$, where A, B, C, D, E, and F are constants.

The equation of the line that passes through the two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $x = A + Bt$, $y = C + Dt$, $z = E + Ft$, where A, B, C, D, E, and F are constants.

The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$. Substituting the given points, we get $y - 3 = \frac{7 - 3}{5 - 2}(x - 2) = \frac{4}{3}(x - 2)$. Simplifying, we get $y - 3 = \frac{4}{3}x - \frac{8}{3} = \frac{4}{3}x - \frac{24}{9} = \frac{4}{3}x - 8$. Therefore, the equation of the line is $y = \frac{4}{3}x - 8 + 3 = \frac{4}{3}x - 5$.

The equation of a line that passes through two points (x1, y1) and (x2, y2) is given by the formula y - y1 = (y2 - y1)/(x2 - x1) * (x - x1). Substituting the values of the two points, we get y - 3 = (7 - 3)/(5 - 2) * (x - 2). Simplifying this equation, we get y = 2x + 1.

The slope of a line passing through two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Substituting the given points, we get: $$m = \frac{7 - 3}{5 - 2}$$ $$m = \frac{4}{3}$$