Problem 23 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(-2,4), m=0$$

Step-by-Step Solution

Answer

The equation of the line that passes through the point (-2,4) and has a slope of 0 is \(y = 4\).

Step 1: Apply the Given Attributes in the Slope-Intercept Formula

Begin by placing the known values into the slope-intercept form \(y=mx+b\). So, you will be replacing \(y\) with 4, \(m\) with 0, \(x\) with -2, resulting in the equation: 4 = 0*(-2) + b. Solve for \(b\).

Step 2: Simplify and Solve for y-intercept

Solving for \(b\), you will get 4 = 0 + b. Therefore, \(b = 4\).

Step 3: Write Final Equation

Finally, rewrite the slope-intercept form with the known values for the slope and y-intercept. The final equation of the line is \(y = 0*x + 4\), which simplifies to \(y = 4\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for writing linear equations. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept.

The slope tells us how steep the line is and in which direction it goes. A positive slope means the line goes upward as it moves from left to right, and a negative slope means it goes downward. The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on a line.

The y-intercept \(b\) is the point where the line crosses the y-axis. At this point, the value of \(x\) is zero. This makes it straightforward to find the y-intercept by simply looking at the constant term in the slope-intercept equation.

When it comes to horizontal lines, they will always have a slope of zero because there is no rise; the line does not go up or down as it moves left to right. Consequently, the slope-intercept form of a horizontal line simplifies to \(y = b\), making the equation depend solely on the y-intercept.

Linear Equations

A linear equation represents a straight line on the coordinate plane. The most general form of a linear equation is \(Ax + By = C\), with \(A\), \(B\), and \(C\) as constants. The slope-intercept form is a specific type of linear equation that makes it easy to graph a line because it directly shows the slope and y-intercept.

Linear equations are characterized by their consistency. The rate of change is constant, which means that the relationship between the x and y values is consistent across the entire line. This is depicted in the equation's straight-line graph.

In addition to the slope-intercept form, linear equations can be written in other formats like point-slope form and standard form. Each form has its own advantages for different types of problems, but the slope-intercept form remains the most intuitive for immediately visualizing the line's characteristics.

Y-Intercept

The y-intercept of a line is a foundational concept in the study of linear equations. It is the point where the line crosses the y-axis of a graph. To find the y-intercept from an equation in slope-intercept form, look for the \(b\) value – it is the numeric term without any \(x\).

In practical scenarios, the y-intercept is often the initial value before any changes have occurred since it represents the output value when \(x\) is zero. It's critical to note that every line has exactly one y-intercept, and for vertical lines, which have an undefined slope, the concept of a y-intercept does not apply since these lines do not cross the y-axis.

When given a graph, the y-intercept is found where the line hits the y-axis, but when starting with an equation, it can be determined algebraically by setting \(x\) to zero and solving for \(y\). This gives us a clear starting point for plotting the line on a graph.