Question

$$y=6(x-9)$$

Short answer

The expanded form of the given equation is \(y = 6x - 54\).

Step-by-step solution

Identify the Exercise

The given exercise is an algebraic equation in the slope-intercept form of a linear equation, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, the equation is \(y = 6(x - 9)\).

Expand the Equation

We need to distribute the multiplication of 6 across the parenthesis. To do so, multiply 6 by each term inside the parenthesis: \(6 \times x\) and \(6 \times -9\). This gives us: \(y = 6x - 54\).

Write the Final Form

The expanded form of the equation is now written without the parenthesis. Thus, the final form of the equation is: \(y = 6x - 54\).

Key concepts

Linear Equation

A linear equation is an algebraic expression that creates a straight line when plotted on a graph. These equations typically have one or two variables and the highest power of the variable is one. In the form of (y = mx + b), m represents the slope of the line, which indicates how steep the line is, and b represents the y-intercept, the point where the line crosses the y-axis. In the context of the exercise (y=6(x-9)), the slope is 6, suggesting that for every one unit increase in x, y increases by 6 units.

In a linear equation, the variables are always to the first power, which means there are no squares or cubes, and no variables are multiplied together. This simplicity allows for quick identification of the slope and intercept, making graphing straightforward. Linear equations are foundational in algebra and are used extensively in various fields, including economics, engineering, and the physical sciences.

Algebraic Equation

An algebraic equation is a mathematical statement where two expressions are set equal to each other, containing one or more unknown variables. The goal is usually to solve for these variables, finding the values that make the equation true. In the exercise given, (y=6(x-9)), we are presented with an algebraic equation that defines a relationship between y and x.

Algebraic equations come in many forms, ranging from simple linear equations to more complex quadratic or polynomial equations. Understanding how to manipulate these equations, through actions like expanding, factoring, or using inverse operations, is crucial for solving them. When we expand equations, we simplify expressions, and we aim to isolate the variable when solving for a particular value. Algebraic equations are not only important in mathematics but also in applying mathematical thinking to solve real-life problems.

Expand Equation

Expanding an equation is an essential algebraic skill that involves extending an expression to remove parentheses or factor out common factors. It requires the application of the distributive property, which in algebra means multiplying a single term by each term within a parenthesis. For example, to expand the given exercise equation (y=6(x-9)), we distribute the 6 across the x and -9.

Here’s the process:

• Multiply 6 by x, which gives us 6x.
• Multiply 6 by -9, resulting in -54.

This results in the expanded form (y = 6x - 54). The purpose of expanding is to simplify the equation for easier manipulation and solving. It can also unveil the slope and y-intercept directly, which is vital for graphing and interpreting the relationship between variables in a linear equation.

Y-Intercept

The y-intercept is a fundamental concept in the graphing of linear equations, representing the point where the line crosses the y-axis on a coordinate plane. It's numerically equal to the value of y when the independent variable, x, is zero. In the slope-intercept form of a linear equation (y = mx + b), the y-intercept is denoted by b.

In the provided exercise, after expanding the equation to (y = 6x - 54), the y-intercept is -54. This tells us that if we were to graph this linear equation, the line would cross the y-axis at the point (0, -54). Identifying the y-intercept is extremely helpful in graphing because it provides a starting point for the line on the graph. Together with the slope, the y-intercept gives all the necessary information to draw the entire line and visualize the relationship between x and y.