Problem 24 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-\frac{3}{5},\) passing through \((10,-4)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 4 = -3/5x + 6\), and in slope-intercept form is \(y = -3/5x + 2\).

Step 1: Find the Point-Slope Form of the Line

For point-slope form, we plug the given values into the equation. The given point is \((10, -4)\), so \(x_1 = 10\) and \(y_1 = -4\). The given slope is \(-3/5\). Plugging these values into the point-slope form equation results in: \(y - (-4) = -3/5 (x - 10)\)

Step 2: Simplify the Point-Slope Equation

We can simplify the equation from step 1 to get: \(y + 4 = -3/5x + 6\)

Step 3: Find the Slope-Intercept Form of the Line

The slope-intercept form of the equation is obtained by rearranging the simplified equation from step 2 in the form \(y = mx + b\), which gives: \(y = -3/5x + 2\)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when working with linear equations. It's an algebraic expression of the form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) indicates the y-intercept, the point where the line crosses the y-axis.

In the context of our textbook problem, the slope-intercept form makes it easy to graph the linear equation quickly. By knowing the slope, \(m = -\frac{3}{5}\), and the y-intercept, \(b = 2\), we can plot the starting point at \(y = 2\) on the y-axis and then use the slope to determine the direction and steepness of the line.

For instance, a negative slope means the line is falling as it moves from left to right. The slope value of \(\frac{3}{5}\) tells us that for every five units we move horizontally to the right, the line will drop by three units downward. \(b\), being positive, indicates the line will cross the y-axis above the origin.

Linear Equations

Linear equations are the backbone of algebra and are used to describe a straight line. They are equations that have variables raised only to the first power and graphically produce a straight line. These equations relate two variables, typically \(x\) and \(y\), in such a way that the pairs of values for these variables describe a line in the Cartesian coordinate system.

The equation from the exercise, \(y = -\frac{3}{5}x + 2\), is a perfect example of a linear equation. It tells us that the relationship between \(x\) and \(y\) is consistent, hence, the graph is a straight line. Furthermore, linear equations can be manipulated into various forms such as the point-slope form, standard form (\(Ax + By = C\)), or the slope-intercept form (\(y = mx + b\)) to suit different purposes like graphing or solving for variables.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the point-slope form equation \(y - y_1 = m(x - x_1)\), \(m\) and \(x_1, y_1\) are constants or coefficients, while \(x\) and \(y\) are the variables. Algebraic expressions become equations when two expressions are set equal to each other, like \(y + 4 = -\frac{3}{5}x + 6\) from Step 2 of the solution.

Improving our understanding of algebraic expressions aids in developing the skill to rearrange these expressions and solve for unknowns – a fundamental aspect of algebra. For instance, converting the point-slope form into slope-intercept form involves simplifying the algebraic expression to solve for \(y\), giving us the ability to identify the slope and y-intercept directly.