Problem 48 - Algebra 1

Question

Write an equation of the line that passes through the two points. $$(-3,-9),(5,7)$$

Step-by-Step Solution

Answer

The equation of a line that passes through the points (-3,-9) and (5,7) is \(y = 2x - 3\).

Step 1: Compute Slope of the Line

Firstly, you need to calculate the slope of the line that passes through the two points. You can use the slope formula, which is \(m = \frac{(y2 - y1)}{(x2 - x1)}\). With the given points being \((-3,-9),(5,7)\), let's plug them into the slope formula. This means you should have: \(m = \frac{(7 - (-9))}{(5 - (-3))}\).

Step 2: Simplify the Slope

By simplification, \(m = \frac{16}{8}=2\). This tells us that the slope of the line is 2.

Step 3: Find the Equation of the Line

Finally, we can use the point-slope form of a line equation that says \(y - y1 = m(x - x1)\), we can use any of two points but for this case, taking the point \((-3,-9)\), will insert this into the equation: \(y - (-9) = 2(x - (-3))\).

Step 4: Simplify the Equation

The above equation simplifies to \(y + 9 = 2(x + 3)\). Distribute the 2 and then subtract 9 from both sides to get \(y = 2x + 6 - 9\). Thus the equation is \(y = 2x - 3\).

Key Concepts

Slope

The slope of a line is a measure of how steep the line is. It tells you how much the line goes up or down for each step it takes to the right. To find the slope between two points, use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where

• \(x_1, y_1\) are the coordinates for the first point,
• \(x_2, y_2\) are the coordinates for the second point.

In our example with points \((-3,-9)\) and \((5,7)\),plug these numbers into the formula:\[m = \frac{7 - (-9)}{5 - (-3)} = \frac{16}{8} = 2\]The slope here is 2, meaning for every unit you move to the right, the line goes up 2 units. This is important for understanding the tilt and direction of your line.

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know the slope and at least one point on the line. This form is handy because it breaks down the necessary information into a simple equation format. The point-slope form equation looks like this:\[y - y_1 = m(x - x_1)\]Here,

• \(m\) stands for the slope,
• \(x_1, y_1\) are the coordinates of a point on the line.

For our line, using the slope \(m = 2\) and the point \((-3,-9)\),plug these into the formula:\[y - (-9) = 2(x - (-3))\]This equation directly relates the change in \(y\) to the change in \(x\) using the known slope, anchored at \((-3, -9)\). The point-slope form easily transitions into other forms when you simplify it.

Simplifying Equations

After writing an equation in point-slope form, the next logical step is to simplify it to make interpretation easier. Simplifying often involves distributing and rearranging terms to convert the equation into slope-intercept form, which looks like\(y = mx + b\).Starting with the equation:\[y + 9 = 2(x + 3)\]Begin by distributing the \(2\) on the right side:\[y + 9 = 2x + 6\]Next, subtract \(9\) from both sides to further isolate \(y\):\[y = 2x + 6 - 9\]Simplify the constants:\[y = 2x - 3\]This is now in slope-intercept form, where the slope \(m\) is \(2\), and the y-intercept \(b\) is \(-3\). Slope-intercept form is widely used because it shows directly how the line behaves, illustrating both its steepness and where it crosses the y-axis.