Problem 15 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(4,-3), m=-2$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \(2x + y = 5\)

Step 1: Write the equation in slope-intercept form

We start with the equation of a line in slope form: \( y = mx + b \). Since we know the slope \( m \) and a point (x, y) through which the line passes, we can substitute these values into the equation to find the y-intercept \( b \). Substituting the given values gives us: \( -3 = -2 \cdot 4 + b \). Solving for \( b \) gives \( b = -3 + 8 = 5 \). The equation of the line in slope-intercept form is thus \( y = -2x + 5 \).

Step 2: Convert equation to standard form

The standard form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) are not both zero. Starting with the slope-intercept form \( y = -2x + 5 \), we can move the \( 2x \) term to the left side of the equation to give \( 2x + y = 5 \). This is now in standard form.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a particularly useful way to represent a line. It follows the formula: \[ y = mx + b \]In this equation, \( m \) represents the slope of the line, while \( b \) is the y-intercept. This form allows you to easily see the slope and the point where the line crosses the y-axis without doing additional calculations.
To create this form, replace \( m \) with the given slope and \( b \) with the y-intercept once it's computed. For example, if a line has a slope of \(-2\) and passes through the point \((4, -3)\), you can substitute those values into the equation. Solve for \( b \) to get your full equation. Here, after solving, you will get \( y = -2x + 5 \).

• Easily determine line characteristics: slope \( m \) and y-intercept \( b \).
• Helps in graphing the line quickly.
• Simply substitute known slope and a point to find unknown elements.

Standard Form

Standard form is another way to represent linear equations, expressed as: \[ Ax + By = C \]In this format, \( A \), \( B \), and \( C \) are integers, with \( A \) and \( B \) not both being zero. Standard form is particularly convenient for certain algebraic operations, such as when solving systems of equations.
Converting from slope-intercept to standard form involves simple algebraic manipulation. From our previous equation \( y = -2x + 5 \), add \( 2x \) to both sides to get \( 2x + y = 5 \). This new equation fits the standard form with \( A = 2 \), \( B = 1 \), and \( C = 5 \).

• Suitable for solving systems of equations by elimination or substitution.
• Can easily identify if the line could be vertical or horizontal.
• Often used in theoretical and analytical proofs.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis. It is an essential value that gives insight into the position of the line on the graph. In the equation \( y = mx + b \), the value of \( b \) is the y-intercept.
Finding the y-intercept is straightforward, especially when you know a point on the line and the slope. From our example, after substituting point \((4, -3)\) into the slope-intercept equation, \( -3 = -2 \cdot 4 + b \), solve for \( b \) to find the y-intercept, \( b = 5 \).

• Visually represents where the line meets the y-axis.
• Provides a fixed point for graphing the line, aiding in drawing accurate linear graphs.
• Key for determining the outset of functions and equations in real-life scenarios.