Problem 49 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the point and has the given slope. $$ (-3,7), m=7 $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the point (-3,7) and has a slope of 7 is \(y = 7x + 28\).

Step 1: Substituting the given values into the slope-intercept equation

First, substitute the given \(x\), \(y\), and \(m\) values into the slope-intercept formula \(y = mx + b\). The values are point (-3,7) and slope \(m\) of 7, thus yielding \(7 = 7(-3) + b\).

Step 2: Solve for the y-intercept \(b\)

Next, solve the equation for \(b\) which results in \(7 = -21 + b\). After simplifying, the solution for \(b\) is found to be \(b = 28\).\]

Step 3: Write the final equation

Finally, use the original slope \(m\) and newly found y-intercept \(b\) to write the final equation in slope-intercept form. Substituting the values into the formula \(y = mx + b\), the equation becomes \(y = 7x + 28\).

Key Concepts

Linear Equations

A linear equation is at the very heart of algebra and serves as the foundation for more complex mathematical concepts. In essence, it represents a straight line when plotted on a graph. The standard form of a linear equation is \( Ax + By = C \), where \(A\), \(B\), and \(C\) denote constants, and \(x\) and \(y\) are variables that represent any point on the line.

The beauty of linear equations lies in their simplicity and the ease with which they can be used to model real-life situations. For instance, suppose you wanted to determine how much money you'd have after saving a certain amount weekly. If that amount stays constant, the relationship between weeks and savings is linear. Understanding and solving linear equations are fundamental skills that will greatly benefit students in their academic and real-life problem-solving.

Slope and Y-Intercept

Knowing the slope and y-intercept of a line is like having a secret recipe that lets you cook up the equation of that line with ease. The slope, often denoted as \(m\), tells you how steep the line is, and it's calculated as the rise over the run between any two points on the line. If you visualize a hill, the slope tells you how quickly you'll ascend or descend as you walk along it.

The y-intercept, signified by the letter \(b\), is where your line crosses the y-axis. It's an essential component because it gives you a starting point for your line. Combining the slope and y-intercept, using the slope-intercept form \(y = mx + b\), provides a complete picture of the line on a graph. It's like having a map that lets you trace the line from its starting point and follow it in the direction indicated by the slope.

Writing Equations of a Line

When writing the equation of a line, the slope-intercept form \(y = mx + b\) is the key tool to translate the numerical information of a line into an algebraic expression. It's straightforward: \(m\) is your slope, the measure of the line's steepness, and \(b\) is your y-intercept, the point where the line hits the y-axis.

Example Application

To illustrate, let's consider the problem where we know a point that the line passes through, \( (-3,7) \), and the slope \(m = 7\). The slope-intercept form makes it easy to plug these values in and find the unknown \(b\):

• First, insert the point into the equation \(7 = 7(-3) + b\).
• Then, solve for \(b\) which gives us \(28\) after simple calculations.
• Finally, write the full equation \(y = 7x + 28\) using the known slope and the calculated y-intercept.

This method ensures that anyone can write the equation of a line as long as they have a point and the slope, simplifying what could otherwise be a complex task into a straightforward formula application.