Problem 106 - College Algebra

Question

Find the value of \(y\) if a linear function goes through the following points and has the following slope: \((10, y),(25,100), m=-5\).

Step-by-Step Solution

Answer

The value of \(y\) is 175.

Step 1: Understand the linear equation formula

The equation of a line in point-slope form is given by \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope of the line.

Step 2: Identify known values

From the problem, the slope \( m = -5 \), and we are given points \((10, y)\) and \((25, 100)\). We will use \((25, 100)\) to find \(y\), considering this point fits the line equation.

Step 3: Write the equation using point-slope form

Substitute the known point \((25, 100)\) and slope \(m = -5\) into the point-slope formula: \( y - 100 = -5(x - 25) \).

Step 4: Express the equation in slope-intercept form

Simplify the equation: \(y - 100 = -5x + 125\). Now solve for \(y\): \(y = -5x + 225\).

Step 5: Use the equation to find unknown \(y\)

Now use the point \((10, y)\): substitute \(x = 10\) into the equation \(y = -5x + 225\). So, \(y = -5(10) + 225\).

Step 6: Solve for \(y\)

Calculate \(-5(10) + 225 = -50 + 225\). Therefore, \(y = 175\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a hugely helpful way to find the equation of a line. It is written as \( y - y_1 = m(x - x_1) \), where:

• \( x_1 \) and \( y_1 \) are the coordinates of a known point on the line.
• \( m \) represents the slope of the line.

This form is very beneficial when we know one point on the line and the slope, just like in our original exercise.To use point-slope form, begin by substituting your known point and slope into the formula. This provides you with a linear equation that's very specific to the line you are considering. By manipulating this equation, you can find other points on the line or rewrite the equation in a different form.

Slope

The slope is a critical concept in linear equations that essentially tells us how 'steep' the line is. Mathematically, the slope \( m \) is defined as the ratio of the change in \( y \) (vertical change) to the change in \( x \) (horizontal change) between two points on the line:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In simpler terms, it tells us how much \( y \) changes for every unit change in \( x \). A positive slope indicates the line rises as \( x \) increases, while a negative slope, like \( -5 \) in our exercise, means the line falls as \( x \) increases.The concept of slope is important because it gives your line direction and can help predict future values in a linear function, which is essential not only in mathematics but also in fields that involve data trends and predictions.

Slope-Intercept Form

Another way to write the equation of a line is using the slope-intercept form, \( y = mx + b \). Here:

• \( m \) represents the slope.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

This form is particularly useful for quickly identifying the slope and y-intercept from the equation itself, making it readily understandable at a glance. Transforming a point-slope equation into the slope-intercept form often simplifies complex problems, as the y-intercept provides a concrete starting point on the graph for drawing the line.In our exercise, by rearranging the point-slope form into slope-intercept form \( y = -5x + 225 \), we could easily substitute \( x \) values to find corresponding \( y \) values and make predictions.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through a coordinate system. In this form of geometry, we use coordinates (usually \((x, y)\)) to express algebraic equations that define lines, curves, and other shapes.The main advantage of using coordinate geometry is its ability to link algebra and geometry, allowing problems to be solved using equations. For example, determining where a line exists in a plane, as done in our exercise where we calculated \( y = 175 \) by substituting an \( x \) value, demonstrates this utility.Practically, coordinate geometry is key for bridging gaps in understanding, as it translates the visual aspects of geometry into algebraic expressions. This crossover enhances problem-solving skills across various mathematical and scientific fields.