Problem 27 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (6,-4),(-1,2) $$

Step-by-Step Solution

Answer

The equation of the line that passes through points (6,-4) and (-1,2) is \( y = -\frac{6}{7}x + \frac{2}{7} \)

Step 1: Plot the Points

Plot the points (6,-4) and (-1,2) on a graph paper.

Step 2: Calculate the Slope

Use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) are the coordinates of the first point (6,-4) and \( (x_2, y_2) \) are the coordinates of the second point (-1,2). So the slope \( m \) equals to \( m = \frac{2 - (-4)}{-1 - 6} = \frac{6}{-7} = -\frac{6}{7} \).

Step 3: Calculate the y-Intercept

Having the slope, now use one of the points and substitute to the form \( y = mx + b \) to find the y-intercept \( b \). Let's use point (6,-4). So we have: \( -4 = -\frac{6}{7}*6 + b \) which simplifies into \( -4 = -\frac{36}{7} + b \). Therefore, the y-intercept \( b \) is \( b = -4 + \frac{36}{7} = \frac{2}{7} \).

Step 4: Write Down the Line Equation

Now we substitute the slope \( m \) and y-intercept \( b \) into the slope-intercept form. The line equation is \( y = -\frac{6}{7}x + \frac{2}{7} \)

Key Concepts

Graphing Points

Plotting points is the first step in understanding how to graph a line on a coordinate system. To graph the points provided in the exercise, you need to locate each point based on its coordinates. Coordinates are written as pairs of numbers:

• The first number is the x-coordinate (horizontal position).
• The second number is the y-coordinate (vertical position).

For instance, the point (6, -4) tells us to move 6 units to the right of the origin (0,0) along the x-axis and then 4 units down along the y-axis. Similarly, the point (-1, 2) means you move 1 unit to the left and 2 units up from the origin. Once both points are plotted on the graph, a line can be drawn through them to represent all the solutions that lie on this line.

Calculating Slope

The slope of a line is a measure of its steepness and direction. It is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]The variables \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of any two points on the line. In this exercise, we use the points \((6, -4)\) and \((-1, 2)\). Plugging these into the formula gives:

• Difference in y-values: \(2 - (-4) = 6\)
• Difference in x-values: \(-1 - 6 = -7\)

This simplifies to the slope \(m = \frac{6}{-7} = -\frac{6}{7}\). This negative slope tells us that the line slants downwards from left to right.

Finding y-Intercept

The y-intercept is the point where the line crosses the y-axis, and it's represented by \(b\) in the slope-intercept equation \(y = mx + b\). To find the y-intercept, use the slope you calculated and one of the points on the line in the equation. Let's use the point (6, -4):\[-4 = -\frac{6}{7}(6) + b\] Simplify:\[-4 = -\frac{36}{7} + b\] Add \(\frac{36}{7}\) to both sides to solve for \(b\):\[b = -4 + \frac{36}{7}\] Convert \(-4\) to \(\frac{-28}{7}\) to easily add fractions:\[b = \frac{-28}{7} + \frac{36}{7} = \frac{2}{7}\]Therefore, the y-intercept is \(\frac{2}{7}\).

Line Equation

The slope-intercept form of a line equation is written as \(y = mx + b\). This form makes it clear what the slope (\(m\)) and the y-intercept (\(b\)) are. Once you have calculated both:

• Slope \(m = -\frac{6}{7}\)
• y-intercept \(b = \frac{2}{7}\)

Substitute these values into the equation:\[y = -\frac{6}{7}x + \frac{2}{7}\]With this equation, you can predict y for any given x, allowing you to understand and analyze the linear relationship represented by the line. This equation is useful for plotting the graph, making predictions, or understanding how changes in x affect y.