Question

Use your answers to Exercise 70 to find the value of \(x\) so that the line passing through the points \((0,3)\) and \((x,-9)\) will have a slope of \(-3\).

Short answer

The value of \(x\) for which the line passing through \((0,3)\) and \((x,-9)\) has a slope of \(-3\) is \(x = 4\).

Step-by-step solution

Identify the given parameters

Identify and note down the given parameters from the question. The known values are \(m = -3\), \(x_1=0\), \(y_1 = 3\), \(x_2 = x\), and \(y_2 = -9\).

Plug in the values into the slope formula

Insert all the known values into the slope formula. Now the equation becomes \(-3 = \frac{-9 - 3}{x - 0}\).

Solve the equation for x

Solve the equation to find the unknown \(x\). Begin by simplifying the right hand side: \(-3 = \frac{-12}{x}\). To isolate \(x\), it's helpful to cross-multiply: -3*x = -12, which simplifies to \(x = 4\) after dividing both sides by -3.

Key concepts

Slope Formula

Understanding the slope formula is crucial when dealing with linear equations and coordinate geometry. The slope of a line measures its steepness and is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two distinct points on the line. The formula is expressed as\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In practical terms, if we have two points, \((x_1, y_1)\) and \((x_2, y_2)\), we subtract the y-coordinate of the first point from the y-coordinate of the second point, and do the same with the x-coordinates. If the slope \(m\) is positive, the line inclines upwards as it moves from left to right. Conversely, a negative slope means the line declines.

For those looking to deeply understand and visualize this, imagine calculating the rise and the run on a set of stairs. The 'rise' corresponds to the vertical change (difference in your y-values), and the 'run' to the horizontal change (difference in your x-values). The steeper the stairs, the greater the 'rise' for each unit of 'run,' resembling a steeper slope.

Linear Equations

Linear equations form the bedrock of coordinate geometry and are essential when plotting straight lines on a graph. A linear equation looks like\[ y = mx + b \]where \(m\) represents the slope and \(b\) is the y-intercept, or the point where the line crosses the y-axis. The beauty of linear equations is in their simplicity as they showcase a direct relationship between two variables: x and y.

When you're given a linear equation, the value of \(y\) depends on the value you choose for \(x\), and the graph of such an equation will always be a straight line. For students solving problems involving linear equations, it's vital to grasp that any solution is in the context of finding the point(s) that adhere to the rule set by the equation. Visual learners may find graphing these equations helpful in understanding their properties, while analytical thinkers may prefer manipulating the algebraic forms.

Coordinate Geometry

Coordinate geometry, or analytic geometry, provides the foundation for plotting and analyzing points, lines, and shapes on the Cartesian plane. Using a pair of numerical coordinates \((x, y)\), we can locate points on this plane and explore the relationships between geometric figures and algebraic equations.

A fundamental concept in coordinate geometry is that of the slope of a line, as it allows us to describe the direction and steepness of the line. Whether you're estimating the gradient of a hill or trying to understand economic growth over time, the concepts overlap abundantly.

To tie it with our initial exercise, the line passing through \((0, 3)\) and \((x, -9)\) is more than just a geometric line; it's also a visual representation of all the points that satisfy the linear equation derived from those two points. In essence, coordinate geometry is the bridge between algebra and visual representation, enhancing our understanding of both.