Problem 118 - College Algebra Essentials

Question

This help you prepare for the material covered in the next section. Write the slope-intercept form of the equation of the line passing through \((-3,1)\) whose slope is the same as the line whose equation is \(y=2 x+1\).

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line is \(y = 2x + 7\).

Step 1: Identify the slope

From the equation \(y = 2x + 1\), we know that the slope \(m\) is 2 as in the general equation, \(y = mx + c\), \(m\) is the coefficient of \(x\).

Step 2: Substitute the point into the equation

To find the intercept \(c\), substitute the point \((-3,1)\) into the equation \(y = mx + c\). Hence, the equation becomes \(1 = 2*(-3) + c\).

Step 3: Solve for the intercept

Solving for \(c\), the equation becomes \(1 = -6 + c\), which simplifies to \(c = 1 + 6 = 7\) when we add 6 on both sides.

Step 4: Write the final equation

Substitute the slope \(m = 2\) and the intercept \(c = 7\) into the equation \(y = mx + c\) to get the final equation of the line, which is \(y = 2x + 7\).

Key Concepts

Linear Equations

Linear equations are at the heart of algebra and are the first type of equations most students encounter that involve both a variable and a constant. A linear equation creates a straight line when graphed on a coordinate plane. The most common form of a linear equation is the slope-intercept form, which is expressed as
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Slope of a Line

The slope of a line is a measure of its steepness, commonly denoted as
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Y-intercept

In linear equations and the graphing of lines, the y-intercept is a key feature. It is the point where the line crosses the y-axis on a coordinate plane. The y-intercept is often denoted as
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Algebraic Problem-Solving

Algebraic problem-solving is a methodical approach to solving equations and understanding the relationships between variables and constants. It involves identifying the unknowns, creating equations, and manipulating these equations to find the values of the unknowns.

Understanding the Problem

The first step in problem-solving is to understand what is asked. In the exercise provided, the task is to find the slope-intercept form of a line with a given slope passing through a specific point.

Developing a Strategy

Next, a strategy is devised which, in this case, includes identifying the slope from a given equation, then using a point on the line to solve for the y-intercept.

Carrying Out the Plan

By substituting the known values into the slope-intercept form equation and solving for the y-intercept, a solution is reached.

Reviewing

Finally, reviewing the steps and the solution ensures understanding and the correct application of algebraic principles. Effective problem-solving in algebra paves the way for tackling more intricate mathematical challenges in advanced subjects like calculus and statistics.