Problem 45 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. $$ 5 x+3 y=3 $$

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = -5/3x + 1\). The slope is \(-5/3\) and y-intercept is \(1\). When graphed, it will be a downward sloping line crossing the y-axis at \(1\).

Step 1: Convert to Slope-Intercept Form

The standard form of an equation, \(Ax + By = C\), can be converted into slope-intercept form, \(y = mx + b\), by solving for y. So for the equation \(5x + 3y = 3\), subtract \(5x\) from both sides to isolate \(y\). This gives you the equation, \(3y = -5x + 3\). To finish solving for \(y\), divide each term in the equation by 3 to get \(y = -5/3x + 1\).

Step 2: Identify Slope and Y-Intercept

In the slope-intercept form equation \(y = mx + b\), \(m\) is the slope and \(b\) is the y-intercept. From the equation \(y = -5/3x + 1\), it can be identified that the slope (\(m\)) is \(-5/3\) and the y-intercept (\(b\)) is \(1\).

Step 3: Graph the Equation

Start by plotting the y-intercept (\(b=1\)) on the y-axis. Since the slope is \(-5/3\), from the y-intercept point, move down 5 units (because the slope is negative) and move to the right 3 units (the denominator of the slope). Draw a line through these points, which will be the graph of the equation.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that describe a straight line when plotted on a graph. These equations are typically written in the form \(Ax + By = C\), known as the standard form. In any linear equation, the variables \(x\) and \(y\) represent coordinates on a graph where a line will be drawn. These equations are fundamental in understanding relationships between two variables, as they show how the value of one variable changes in relation to a change in another variable.

In the process of solving linear equations, often we convert them into another form called the slope-intercept form. This transformation allows for easier graphing and interpretation of the line's characteristics, as it explicitly reveals the slope and y-intercept of the line. This makes it easier to visualize how the line behaves within a coordinate system.

Graphing Linear Equations

Graphing equations is an essential skill in mathematics that allows you to visually understand the relationship between variables. With any linear equation expressed in slope-intercept form \(y = mx + b\), you need to follow a couple of steps to accurately graph it:

• First, identify the y-intercept, \(b\), which is the point where the line crosses the y-axis. This is where you will start plotting your line.
• Next, use the slope, \(m\), which tells you how steep the line is and in what direction it moves. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.

In our example, after converting the equation \(5x + 3y = 3\) to \(y = -5/3x + 1\), we start plotting by marking the y-intercept at \(y = 1\). Utilizing the slope \(-5/3\), indicates a descent of 5 units vertically with each 3 units it moves horizontally to the right. This sequence of operations provides a clear and systematic method for drawing the line that represents the equation.

Slope and Y-Intercept Defined

The slope and y-intercept are two key characteristics of the line that a linear equation represents. They help provide a clear understanding of how the line interacts with the coordinate system.

• The slope \(m\), expressed as a fraction like \(-5/3\), describes the line's steepness and direction. A slope of \(-5/3\) means that for every 3 units you move right on the x-axis, the line moves down 5 units on the y-axis. The negative sign indicates a downward trend.
• The y-intercept \(b\) is simply where the line crosses the y-axis. In this case, the intercept is \(1\). This tells us that when \(x = 0\), \(y\) has a value of \(1\).

Understanding these components is crucial for interpreting and graphing the equation effectively. Together, they provide a complete picture of the line's position and angle, offering insight into the linear relationship it defines.