Problem 13 - Algebra 1

Question

Write an equation of the line that passes through the given points. $$ (-4,-1),(-9,2) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-4,-1) and (-9,2) is y = -0.6x - 3.4.

Step 1: Calculate the slope

The slope of the line that passes through points (-4,-1) and (-9,2) can be found using the formula m = (y2 - y1) / (x2 - x1). Substitute in the given points to get m = (2- -1) / (-9 - -4) = 3/-5 = -0.6.

Step 2: Substitute slope and one point in equation y=mx+b to find y-intercept

Substitute the calculated slope and one of the points, say (-4,-1), into the equation y = mx + b. This gives -1 = -0.6 * -4 + b. Solve for b to get b = -1 - (-0.6*-4) = -1 - 2.4 = -3.4.

Step 3: Write the equation of the line

Finally, having calculated the slope and the y-intercept, write the equation of the line as y = mx + b, which in this case is y = -0.6x -3.4.

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is, which directly corresponds to how much the y-value changes for a given change in the x-value. It can be thought of as the line's 'tilt'. The steeper the line, the greater the slope. Mathematically, slope is represented by the letter 'm' and can be calculated when you have two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \), using the formula: \[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\].

For instance, given two points \( (-4, -1) \) and \( (-9, 2) \), the slope is found as \(m = \frac{{2 - (-1)}}{{-9 - (-4)}} = \frac{{3}}{{-5}} = -0.6\). This negative slope suggests that the line is descending from left to right, indicative of a negative relationship between the x and y variables.

Understanding the slope is crucial as it dictates the direction and angle of the line on a graph, and ultimately, the relationship between x and y.

Point-Slope Form

Point-slope form is an algebraic equation used to express the equation of a line when you know the slope and at least one point the line passes through. The general form of this equation is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the known point.

Once the slope is calculated, as done in our example where \( m = -0.6 \), you can substitute the slope and coordinates of one of the points into the equation. If we use the point \( (-4, -1) \) with our slope, the point-slope form looks like \( y - (-1) = -0.6(x - (-4)) \), which simplifies to \( y + 1 = -0.6(x + 4) \).

This form is particularly handy for creating equations of lines quickly and is an essential tool for algebra students. It's also a stepping stone to rearranging the equation into slope-intercept form, which is the most commonly used form for graphing linear equations.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis. In the coordinate plane, this is where \( x = 0 \). In terms of an equation, the y-intercept is represented by 'b' in the slope-intercept form of a linear equation \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept.

Finding the y-intercept involves rearranging the point-slope form or substituting into the slope-intercept form and solving for 'b' with a known point. In our exercise, after substituting the slope \( m = -0.6 \) and the point \( (-4, -1) \) into the formula, the calculation \( -1 = -0.6 \times -4 + b \) gives us the y-intercept \( b = -3.4 \). This lets us write the equation of the line with both slope and y-intercept as \( y = -0.6x - 3.4 \).

The y-intercept is a critical piece of information because it gives a starting point for plotting the line on a graph and is also the 'initial value' in many applied mathematics problems, representing the starting condition when all other variables are zero.