Problem 14 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in slope-intercept form. $$y-4 x=9$$

Step-by-Step Solution

Answer

The slope-intercept form of the given equation is \( y = 4x + 9 \)

Step 1: Rearrange the Given Equation

Firstly, the equation is rearranged to isolate \( y \). This is done by adding \( 4x \) to both sides of the equation. So, \( y = 4x + 9 \)

Step 2: The converted form

The rearranged equation is now in the required slope-intercept form

Key Concepts

Linear Equations

Understanding linear equations is crucial to mastering algebra and various mathematical applications. In essence, a linear equation creates a straight line when plotted on a graph. The beauty of these equations lies in their simplicity; they have one or more variables with no exponents or powers higher than one.

The general form of a linear equation in two variables, typically x and y, is expressed with the equation: \( Ax + By = C \), where A, B, and C are real numbers, and A and B are not both zero. Here, the values of 'x' and 'y' can vary, giving an infinite number of solutions that form a line on the Cartesian plane.

The slope-intercept form is a variation of this, specially structured as \( y = mx + b \). Here, 'm' represents the slope of the line, indicating its steepness and direction, while 'b' is the y-intercept, showing where the line crosses the y-axis. This form is particularly useful because it makes finding the slope and y-intercept of a line direct and unambiguous.

Algebraic Manipulation

Algebraic manipulation refers to the application of various arithmetic operations and algebraic properties to reformulate expressions or equations. Its goal is to simplify complex algebraic expressions or rearrange equations to make them more tractable.

When dealing with linear equations, this manipulation often involves:

• Adding or subtracting the same number or expression from both sides of the equation to maintain equality.
• Multiplying or dividing both sides of the equation by a non-zero number.
• Using distributive properties to remove parentheses.
• Combining like terms to simplify the equation further.

With these tools, we can transform equations into more usable forms — for example, taking an equation from \( Ax + By = C \) and converting it to the slope-intercept form, which is simpler to use in many problems.

Equation Rearrangement

Equation rearrangement is a specific type of algebraic manipulation that focuses on rearranging the structure of an equation to isolate a particular variable. This process is incredibly important for solving equations and graphing linear equations.

For example, to isolate 'y' in the equation \( Ax + By = C \), we would perform the following steps:

• Subtract Ax from both sides to get \( By = C - Ax \).
• Divide both sides by B, the coefficient of y, assuming B is not zero, to achieve the form \( y = \frac{C}{B} - \frac{A}{B}x \).
Once 'y' has been isolated, the equation takes on the slope-intercept form, thereby exposing the slope and y-intercept directly. Rearranging equations is not just a mechanical process; it requires an understanding of the properties of equality and the goals we aim to achieve, whether solving for a variable or preparing to graph.