Question

Write the equation of each straight line in slope-intercept form, and make a graph. Slope \(=-1.5 ; y\) intercept \(=3.7\)

Short answer

Equation of the line: y = -1.5x + 3.7.

Step-by-step solution

Identify Slope and Y-Intercept

From the given information, the slope of the line (m) is -1.5, and the Y-intercept (b) is 3.7. Slope-intercept form of a line is given by the equation y = mx + b.

Write the Equation

Substitute the given slope and Y-intercept into the slope-intercept form. The slope (m) is -1.5 and Y-intercept (b) is 3.7, so the equation of the line is y = -1.5x + 3.7.

Plot the Y-Intercept

Start by plotting the Y-intercept (0, 3.7) on the graph. This is the point where the line crosses the Y-axis.

Use the Slope to Plot the Second Point

From the Y-intercept, use the slope to determine the second point. Since the slope is -1.5, it means for every 1 unit you move horizontally to the right, you move 1.5 units down. You can choose a point to the right of the Y-intercept for this purpose, for example, from the point (0, 3.7), you could move to the point (1, 3.7 - 1.5) = (1, 2.2).

Draw the Line

Connect the two points with a straight line and extend it on both sides, making sure it has the correct slope.

Key concepts

Linear Equations

Understanding linear equations is vital for solving problems that involve finding the equation of a line. A linear equation represents a straight line when plotted on a graph. It relates two variables, usually x and y, in the form of an equation like \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.

The slope tells us how steep the line is, while the y-intercept informs us where the line crosses the y-axis. When we're given certain values for these parameters, as in the originally provided exercise with a slope of \( -1.5 \) and a y-intercept of \( 3.7 \), we simply plug these into the linear equation formula to find the equation of the line. Therefore, in this case the equation of the line is \( y = -1.5x + 3.7 \).

It's important to grasp that linear equations yield straight lines and that they come in various forms, including slope-intercept form, point-slope form, and standard form. However, the slope-intercept form is the most commonly used because it directly reveals the slope and y-intercept, making graphing relatively straightforward.

Graphing Straight Lines

Graphing straight lines may seem challenging at first, but it is essentially about plotting points and connecting them in a way that accurately represents the equation of the line.

Starting with the equation in slope-intercept form, we first plot the y-intercept on the graph, as it's the point where the line intersects the y-axis. From there, we use the slope to find additional points on the line. The slope \(m\) gives us the direction and steepness of the line: a positive slope means the line rises from left to right, while a negative slope indicates it falls.

In the solution provided, after plotting the y-intercept \( (0, 3.7) \), we follow the slope of \( -1.5 \) to find a second point. We move 1 unit to the right (along the x-axis) and 1.5 units down (since it's a negative slope). This leads us to the point \( (1, 2.2) \). Plotting these two points and drawing a line through them gives us the graph of the line. To ensure our line is accurate, we can check other points on the line or use a ruler or a straightedge to aid in drawing a precise straight line.

Slope and Intercept

The slope and intercept are foundational concepts in understanding the graph of a linear equation. The slope, often denoted by \(m\), indicates the rate at which y changes with respect to x. It can be considered the 'tilt' or 'inclination' of the line. Usually expressed as a fraction like \(\frac{rise}{run}\), it shows how many units the line goes up or down (rise) for each unit it moves to the right (run).

The intercepts are where the line crosses the axes. The y-intercept (usually denoted by \(b\)) is the point where the line meets the y-axis when x is zero. For the x-intercept, it’s the point where the line crosses the x-axis, which is found by setting y to zero and solving for x.

Important Considerations

In our exercise example, the negative slope of \( -1.5 \) indicates the line falls as it moves from left to right, and the y-intercept of \( 3.7 \) places the starting point above the origin. Students should note that the x-intercept could also be determined, but in the provided example, only the y-intercept is required for the solution. Moreover, the slope being a fraction sometimes requires additional steps to accurately plot the second point, but a solid understanding of these concepts will enable students to graph the line with precision.