Problem 12 - Algebra 1

Question

Write an equation of the line that passes through the given points. $$ (5,12),(6,-2) $$

Step-by-Step Solution

Answer

The equation of the line that passes through points (5,12) and (6,-2) is y = -14x + 82.

Step 1: Calculate the Slope of the Line

Use the formula m = (y2 - y1)/(x2 - x1) to calculate the slope of the line. Here, (x1, y1) is (5, 12) and (x2, y2) is (6, -2). After substitution, we get m = (-2 - 12)/(6 - 5), which simplifies to m = -14.

Step 2: Substitute the Slope and One Point into the Point-Slope Form of the Line Equation

Now that we have the slope m= -14, we will use this along with one of the points it passes through into the point-slope form of the line equation (y - y1 = m(x - x1)). If we use point (5,12), we get \(y - 12 = -14(x - 5)\).

Step 3: Solve the Equation

Simplify the above equation to get the equation of the line in slope-intercept form. Distribute -14 to get \(y - 12 = -14x + 70\). Adding 12 to both sides gives the linear equation y = -14x + 82.

Key Concepts

Slope Calculation

To understand how to write an equation of a line, we start by calculating its slope. The slope of a line indicates its steepness and direction. We use the slope formula to determine this value: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this formula, - \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line. - \(m\) represents the slope.For our exercise, the points given are \((5, 12)\) and \((6, -2)\). Plugging these points into the formula gives:\[ m = \frac{-2 - 12}{6 - 5} = \frac{-14}{1} = -14 \]This result tells us that the line moves downward with a steepness of 14 units downward for every unit it moves to the right. It's a negative slope, which slope means the line decreases as it moves from left to right.

Point-Slope Form

Once we've determined the slope, we can use the **Point-Slope Form** of a line equation, useful for writing the equation of a line with a known slope and a point on the line. The formula is:\[ y - y_1 = m(x - x_1) \] Where:- \(m\) is the slope, - \((x_1, y_1)\) is a point on the line.Using the slope calculated as -14 and selecting point \((5, 12)\), we substitute into the formula:\[ y - 12 = -14(x - 5) \]This equation represents the line in its point-slope form. It shows how the y-coordinate changes as the x-coordinate changes, based on the slope value. This form is particularly useful for graphing because it gives you direct insight into the line's angle and position.

Slope-Intercept Form

The last step is to convert the equation from point-slope to slope-intercept form, which is \[ y = mx + b \]This compact form makes it easy to graph the line by identifying the slope \(m\) and the y-intercept \(b\), where the line crosses the y-axis.Starting from the previous point-slope equation:\[ y - 12 = -14(x - 5) \]We can rearrange it by distributing and simplifying:1. Distribute \(-14\) - \(y - 12 = -14x + 70\)2. Add 12 to both sides - \(y = -14x + 70 + 12\)3. Simplify further to find the equation - \(y = -14x + 82\)Now the equation is in the slope-intercept form where- The slope \(m\) is \(-14\),- The y-intercept \(b\) is \(82\), indicating the line crosses the y-axis at \((0, 82)\). This form is quick and easy to interpret if you need to represent or analyze the relationship visually or graphically.