Problem 21 - Algebra 1

Question

Find the slope and the y-intercept of the graph of the equation. $$ 12 x+4 y-2=0 $$

Step-by-Step Solution

Answer

The slope of the given line is -3 and the y-intercept is 0.5.

Step 1: Rewrite the equation in the form y = mx + b

To rewrite the equation, isolate y on one side to get: \(y = -\frac{12}{4}x + \frac{2}{4}\) which simplifies to : \(y = -3x + 0.5\)

Step 2: Identify the slope

From the equation \(y = -3x + 0.5\), the coefficient of x is the slope. Hence, the slope, \(m = -3\).

Step 3: Identify the y-intercept

From the equation \(y = -3x + 0.5\), the constant term is the y-intercept. Hence, the y-intercept, \(b = 0.5\).

Key Concepts

Linear Equations

Linear equations are the foundation of algebra and are fundamental in understanding how to investigate relationships between two variables. They appear in the form ax + by = c, where a, b, and c are constants. The graphs of these equations are straight lines. When solving for these types of problems, it is usually best to rearrange the equation into a more usable form, such as the slope-intercept form. This not only helps in graphing the line but also in identifying key features like the slope and y-intercept.

In the given exercise, the equation provided was initially in standard form. By manipulating it algebraically, we can isolate y and convert it into slope-intercept form to find the slope and y-intercept easily.

Graphing Linear Equations

When it comes to graphing linear equations, understanding the slope and the y-intercept is crucial. The graph of any linear equation will be a straight line. The slope, often represented by m, tells us how steep the line is, while the y-intercept, represented by b, tells us where the line crosses the y-axis. To graph a linear equation, one can start at the y-intercept and then use the slope to find other points on the line. For instance, with a slope of -3, one would move down 3 units on the y-axis for every 1 unit moved to the right on the x-axis from the y-intercept. These points can then be connected to draw the line.

Solving for y

Solving for y is a pivotal skill when working with linear equations. This process allows us to express y as a function f(x), which is crucial for graphing and understanding the equation's behavior. The objective is to rearrange the equation to isolate y on one side, resulting in the form y = mx + b. Algebraic manipulation involving adding, subtracting, multiplying, or dividing both sides of the equation by the same amount helps achieve this. In the exercise provided, dividing by 4 and then rearranging the terms gave us y in terms of x, making it clear what the slope and y-intercept are for the equation.

Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is particularly useful because it gives direct insight into the characteristics of the graph without requiring additional computation. The slope indicates the direction and steepness of the line, and the y-intercept specifies the point where the graph crosses the y-axis. In our problem, after simplifying the original equation to y = -3x + 0.5, it becomes apparent that the slope m is -3, indicating a line that decreases from left to right. The y-intercept b is 0.5, pinpointing the exact location on the y-axis where the line begins.