Problem 46 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Slope \(-\frac{3}{5} ;\) through (4,-1)

Step-by-Step Solution

Answer

3x + 5y = 7

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is a fundamental equation in coordinate geometry, represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line. In this context, the slope \( m \) is given as \( -\frac{3}{5} \), and the line passes through the point \((4, -1)\). First, we aim to insert these known values into the slope-intercept form to calculate \( b \).

Step 2: Substitute for Slope and Coordinates to Find y-intercept

Begin by substituting the slope \( m = -\frac{3}{5} \) and the coordinates of the point \((4, -1)\) into the slope-intercept form \( y = mx + b \). This helps to find the y-intercept, \( b \): \[ -1 = -\frac{3}{5}(4) + b \] \[ -1 = -\frac{12}{5} + b \] \[ b = -1 + \frac{12}{5} \] By converting -1 into a fraction with denominator 5, \(-\frac{5}{5}\), we get:\[ b = -\frac{5}{5} + \frac{12}{5} = \frac{7}{5} \]

Step 3: Write the Equation in Slope-Intercept Form

Substitute the found value of \( b \) back into the slope-intercept form of the equation:\[ y = -\frac{3}{5}x + \frac{7}{5} \] This gives the equation of the line in slope-intercept form.

Step 4: Convert to Standard Form

The standard form of a linear equation is represented as \( Ax + By = C \) where \( A, B, \) and \( C \) are integers, and \( A \) is positive. Convert the slope-intercept form \( y = -\frac{3}{5}x + \frac{7}{5} \) into standard form by eliminating the fractions:Multiply every term by 5 to clear the fraction:\[ 5y = -3x + 7 \] Rearranging terms gives:\[ 3x + 5y = 7 \] This is the equation in standard form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express a linear equation which is both straightforward and highly useful in various mathematical problems. It is given by the equation \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating its steepness or incline.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In our problem, the given slope is \(-\frac{3}{5}\) and the line passes through the point \((4, -1)\). By substituting the x and y values into the equation, we solve for \( b \), the y-intercept. This process makes it clear how changes in \( m \) and \( b \) affect the line's behavior and placement on the graph. Breaking down this equation helps understand how a change in either the slope or intercept shifts the entire line. Remember, calculating the y-intercept first simplifies the task of writing equations of lines.

Standard Form

Writing linear equations in standard form \( Ax + By = C \) allows for a neat representation, especially useful in more advanced algebra and when working with systems of equations. In this form:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) is non-negative, and ideally, should be positive.

To convert from slope-intercept form to standard form, we aim to clear any fractions and ensure the equation is arranged in a particular order (with \( x \) and \( y \) on one side and constants on the other). For our example, the slope-intercept form \( y = -\frac{3}{5}x + \frac{7}{5} \) is converted into standard form by multiplying through by 5 to eliminate fractions, then rearranging to yield \( 3x + 5y = 7 \). This method solidifies the consistency of variable coefficients when graphing or solving, enabling precise work with systems or in geometric contexts.

Coordinate Geometry

Coordinate geometry, sometimes called analytic geometry, involves visualizing geometric shapes and lines within a coordinate plane, using numerical approaches. Here, we use it to analyze a line given a point and slope:

• Involves plotting points, lines, and curves using coordinates.
• Utilizes algebraic techniques to figure out geometric properties like the slope, distance between points, or intersection points.

The example provided shows coordinate geometry's strength by translating a real-world situation into a graph. Given that the line must pass through \((4, -1)\) with a slope of \(-\frac{3}{5}\), our task is to find out the equation of this line, both in slope-intercept and standard form. This equation becomes a tool for identifying any other points on the line or understanding how it interacts with other lines. Coordinate geometry thus bridges numerical methods and spatial visualization, emphasizing their efficacy in solving practical problems.