Geometry: Parallel and Perpendicular Lines

The lines are neither parallel nor perpendicular.

The equation is $y = 3x - 5$.

The slope-intercept form is $y = \frac{2}{3}x - 2$.

<p>The slope is 150.</p> Signup and view all the answers

<p>The solution is (3, 0).</p> Signup and view all the answers

<p>If the product of their slopes is -1, then the lines are perpendicular.</p> Signup and view all the answers

<p>-2</p> Signup and view all the answers

<p>Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the points.</p> Signup and view all the answers

<p>2</p> Signup and view all the answers

<p>First, graph the line $y = 3x - 2$ as a dashed line, then shade below the line.</p> Signup and view all the answers

<p>The slope indicates the rate of change of the y-value for every unit change in the x-value, showing the steepness and direction of the line.</p> Signup and view all the answers

<p>The slopes of two parallel lines must be equal.</p> Signup and view all the answers

<p>The slope can be calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points.</p> Signup and view all the answers

<p>You can use the number of vehicles and the total related to the cost or capacity to set up equations based on the relationships described.</p> Signup and view all the answers

<p>To convert to slope-intercept form, solve for y to isolate it, resulting in the format $y = mx + b$, where m is the slope.</p> Signup and view all the answers

<p>The slope represents the rate of change between the two variables, indicating how much one variable changes for a unit change in the other.</p> Signup and view all the answers

<p>You can use graphing, substitution, or elimination. The choice depends on the complexity of the equations and personal preference for visual versus algebraic solutions.</p> Signup and view all the answers

<p>Lines are parallel if they have the same slope and perpendicular if their slopes are negative reciprocals of each other.</p> Signup and view all the answers

<p>To convert, isolate y; for example, from $2x - 3y = 6$, rearranging gives $y = rac{2}{3}x - 2$.</p> Signup and view all the answers

<p>Set up an equation based on total capacity in relation to the number of vehicles; for instance, $C = mv + nb$, where m is capacity per van and n is capacity per bus.</p> Signup and view all the answers

<p>They are perpendicular because the product of their slopes is -1.</p> Signup and view all the answers

<p>The slope is the coefficient of x, which is 4.</p> Signup and view all the answers

<p>The slope of 5 indicates that for each additional vehicle rented, there are 5 more students.</p> Signup and view all the answers

<p>The slope of the perpendicular line is -0.5, so the equation is $y + 2 = -0.5(x - 3)$, which simplifies to $y = -0.5x - 0.5$.</p> Signup and view all the answers

<p>The slope is -0.75.</p> Signup and view all the answers

<p>-5</p> Signup and view all the answers

<p>2</p> Signup and view all the answers

<p>y = 2x + 1</p> Signup and view all the answers

<p>-2</p> Signup and view all the answers

<p>-1.5</p> Signup and view all the answers

<p>You can find the slope by choosing two points on the line, calculating the change in y-values over the change in x-values, or using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.</p> Signup and view all the answers

<p>Use the formula $m = \frac{8 - 4}{7 - 3} = 1$.</p> Signup and view all the answers

<p>The slope of the perpendicular line would be $-\frac{1}{3}$, since perpendicular slopes are negative reciprocals.</p> Signup and view all the answers

<p>First, identify the slope of the parallel line; then use the point-slope form $y - y_1 = m(x - x_1)$ to write the equation.</p> Signup and view all the answers

<p>Slope is the ratio of the vertical change to the horizontal change between any two points on the line, often denoted as $m$.</p> Signup and view all the answers

<p>Rearrange the equation to isolate $y$ on one side, resulting in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.</p> Signup and view all the answers

<p>Two lines are parallel if they have the same slope; check the coefficients of $x$ in their equations after converting to slope-intercept form.</p> Signup and view all the answers

<p>The slope is $-2$, indicating that the line descends at a rate of 2 units down for every unit it moves to the right.</p> Signup and view all the answers

<p>Identify the variables and their rates of change, then translate this into a linear equation, often in the form $y = mx + b$.</p> Signup and view all the answers

<p>First, find the slope of the new line, which is -1, then use point-slope form: $y + 1 = -1(x - 4)$.</p> Signup and view all the answers