Question

Write the equation in slope-intercept form. Then graph the equation. $$ 2 x-y+3=0 $$

Short answer

The equation in slope-intercept form is \(y = 2x + 3\). The slope is 2 and y-intercept is 3. After plotting these on the graph, a line is obtained.

Step-by-step solution

Rewrite the equation in slope-intercept format (y = mx + b)

The given equation is \(2x - y + 3 = 0\). To write this in slope-intercept form, solve for y. The result is \(y = 2x + 3\)

Identify the slope and y-intercept

From the equation \(y = 2x + 3\), it is noticeable that the slope (m) is 2 and the y-intercept (b) is 3.

Graph the line

With the slope, 2, rise/run is 2/1, so from the y-intercept (0,3), move two units up and one unit to the right for the next point (1,5). Draw a line through these two points to complete the graph.

Key concepts

Graphing Linear Equations

Graphing linear equations is a foundational technique in algebra that allows students to visually represent solutions to linear equations on a coordinate plane. A linear equation, such as the one provided in the exercise, can always be written in the form of y = mx + b, where m is the slope and b is the y-intercept. To graph a linear equation, we simply plot the y-intercept on the y-axis, and then use the slope to determine the direction and steepness of the line.

For our example (y = 2x + 3), the y-intercept is the point where the line crosses the y-axis, which occurs at (0,3). Once the y-intercept is plotted, the slope tells us how to find the next point. For every one unit we go to the right along the x-axis, the slope tells us to go up by two units, since our slope m is 2. Repeating this process and connecting the points with a straight line completes the graph of the equation.

Slope and Y-intercept

Slope and y-intercept are essential concepts that define the characteristics of a linear equation. The slope, represented by m in the slope-intercept form y = mx + b, indicates the rate at which y changes relative to x. In simple words, it represents how 'steep' a line is. A positive slope means the line ascends from left to right, while a negative slope means it descends.

The y-intercept b points out the exact location where the line crosses the y-axis. It is the value of y when x is zero. In the exercise, the slope of the line is 2, suggesting that for every 1 unit increase in x, y increases by 2 units. Conversely, the y-intercept is 3, indicating the line crosses the y-axis at (0,3). Understanding these components is crucial for graphing the equation correctly.

Equation Solving

Equation solving involves finding the values for the variables that make the equation true. It is a fundamental skill in algebra, and there are many methods to solve equations, depending on their complexity. For the slope-intercept form, the goal is to isolate y on one side of the equation to easily identify the slope and intercept. In our example, the process was starting with 2x - y + 3 = 0 and rearranging it to y = 2x + 3 by moving terms and following basic algebraic principles.

Equation solving often requires manipulation of terms, including adding, subtracting, multiplying, or dividing both sides of the equation to achieve a certain form. In the context of graphing, obtaining the slope-intercept form unlocks the ability to quickly and effectively graph a linear equation and understand its geometric interpretation on a graph.