Problem 39 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ x+y=0 $$

Step-by-Step Solution

Answer

The slope-intercept form of the given equation is \(y = -x\). The graph is a straight line passing through the origin (0,0) and sloping downwards from left to right.

Step 1: Convert to Slope-Intercept Form

To convert the equation to slope-intercept form, isolate \(y\). Subtract \(x\) from both sides of the equation. The result is \(y = -x\). Therefore, the slope-intercept form of the equation is \(y = -x\).

Step 2: Identify the Slope and the y-intercept

The slope of the line \(m\) is the coefficient of \(x\) in the equation. In this case, \(m = -1\). The y-intercept \(b\) is the constant term in the equation. In this case, \(b = 0\). The y-intercept is the point where the line crosses the y-axis.

Step 3: Graph the Equation

To graph the equation, first plot the y-intercept, which is at the point (0,0). Then, since the slope is -1, for every 1 unit increase in \(x\) (moving to the right on the x-axis), decrease \(y\) by 1 unit (moving down on the y-axis). Plot this point and draw a straight line that passes through both points.

Key Concepts

Graphing Equations

Graphing equations allows us to visualize mathematical relationships on a coordinate plane. When we graph linear equations, we look to plot points that represent solutions to the equation. These points usually form a straight line, hence the name "linear."

• We begin by identifying key components, such as the slope and y-intercept, from the equation in slope-intercept form.
• The initial point we plot is the y-intercept, where the line crosses the y-axis.
• From this intercept, we utilize the slope to determine the position of another point on the graph.
• Each increase in the x-coordinate by 1 unit results in a corresponding change in the y-coordinate determined by the slope.

Graphing is an important skill used in many fields, aiding in visual comprehension and analysis of data.

Linear Equations

Linear equations are algebraic expressions that represent straight-line relationships between variables. They are typically written in the form \(ax + by = c\). Solving these equations often involves finding pairs of values that satisfy the equation, which form points on a graph.

• In two variables, linear equations describe lines in a plane.
• When we manipulate linear equations into slope-intercept form, \(y = mx + b\), we simplify graphing by easily identifying slope \(m\) and y-intercept \(b\).
• These equations can be solved algebraically by finding the value of one variable in terms of others, or graphically by representing them as lines and determining intersections.

Understanding linear equations is a fundamental aspect of algebra and is integral to studying higher mathematics, providing a basis for understanding more complex relationships.

Slope and Y-Intercept

The slope and y-intercept are crucial components in understanding and graphing linear equations.

• The slope \(m\) indicates the steepness of a line and its direction of increasing or decreasing. It is calculated as "rise over run," or the change in \(y\) divided by the change in \(x\).
• A positive slope means the line ascends from left to right, while a negative slope indicates it descends.
• The y-intercept \(b\) is the point at which the line crosses the y-axis. It represents \(y\) when \(x\) is zero and acts as a starting point for graphing.
• By knowing both the slope and y-intercept, one can accurately draw the graph of an equation by plotting the intercept and using the slope for direction.

These concepts are essential tools in analyzing linear relationships graphically and algebraically, providing insight into how variables relate to each other.