Problem 21

Question

Write the slope-intercept equation of the line that passes through the two given points. $$ (2,7),(3,10) $$

Step-by-Step Solution

Verified

Answer

The equation of the line is $ y = 3x + 1 $.

1Step 1: Identify the Formula

The first step is to understand which formula we need to use. The slope-intercept form of a line is $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

2Step 2: Calculate the Slope

The slope $ m $ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated by the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $. Substitute the values from the points (2, 7) and (3, 10): $ m = \frac{10 - 7}{3 - 2} = 3 $.

3Step 3: Use the Point-Slope Formula

With the slope $ m = 3 $, we can use any of the given points to find $ b $ using the equation $ y = mx + b $. Let's use point $(2,7)$: $ 7 = 3(2) + b $.

4Step 4: Solve for the Y-Intercept

Solve the equation $ 7 = 6 + b $ to get $ b = 1 $.

5Step 5: Write the Equation

Now that we have both the slope $ m $ and the y-intercept $ b $, the equation of the line is: $ y = 3x + 1 $.

Key Concepts

Slope CalculationY-InterceptLinear Equations

Slope Calculation

The slope of a line is a measure of how steep the line is. It's often referred to as the "rise over run," describing how much the line goes up or down for each unit it moves horizontally. Calculating the slope between two points, such as

(2, 7), and
(3, 10),

can be done using the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $. Here,

$ y_2 = 10 $,
$ y_1 = 7 $,
$ x_2 = 3 $, and
$ x_1 = 2 $.

Plug these values into the formula to find the slope: \[ m = \frac{10 - 7}{3 - 2} = \frac{3}{1} = 3. \]This means the line rises 3 units for every 1 unit it moves to the right, showing a positive correlation between the x and y values.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. At this point, the x-coordinate is zero. To find the y-intercept $ b $ in the equation $ y = mx + b $, you can use a known point and the slope.
For example, with the point

(2, 7),

which we used in our slope calculation, we substitute back into the equation: \[ 7 = 3(2) + b \]This simplifies to: \[ 7 = 6 + b \]Solving for $ b $, we subtract 6 from both sides: \[ b = 1. \]Hence, the y-intercept, $ b $, is 1. This means the line crosses the y-axis at (0,1). It's the starting point of the line on the y-axis when $ x = 0 $.

Linear Equations

Linear equations describe straight lines and can often be found in the slope-intercept form $ y = mx + b $. This representation is straightforward due to its clear components:

$ m $, the slope, indicating how steep the line is,
and $ b $, the y-intercept, indicating where the line crosses the y-axis.

Incorporating the values found in the previous steps, the equation of our line becomes: \[ y = 3x + 1. \]A linear equation like this visually depicts the relationship between two variables, $ x $ and $ y $, showing that for each unit increase in $ x $, we see the corresponding increase in $ y $ determined by the slope. Linear equations are fundamental in algebraic studies for modelling and solving real-world problems by predicting outcomes based on consistent rates of change.

Problem 21

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