Question

Find the slope and the y-intercept of the graph of the equation. $$ y=2 x+1 $$

Short answer

The slope of the line is 2 and the y-intercept is 1.

Step-by-step solution

Identify the Slope

Look at the coefficient of \(x\) in the equation \(y = 2x + 1\). Here, the coefficient of \(x\) is '2', which is the slope of the line, \(m\).

Identify the Y-intercept

The y-intercept is the constant term in the equation. Here, the constant term is '1'. Thus, \(b = 1\), which is the point where the line crosses the y-axis.

Key concepts

Linear Equations

When we talk about linear equations, we're referring to a fundamental algebraic expression that allows you to find a straight line when graphed. Take, for example, the equation
\(y = 2x + 1\).
This is a prototype of a linear equation in the form
\(y = mx + b\),
where \(m\) represents the slope, which indicates how steep the line is, and \(b\) signifies the y-intercept, the point where the line crosses the y-axis. A linear equation always produces a straight line, and the relationship between the two variables, \(x\) and \(y\), is proportional.

Graphing Lines

Graphing lines is an essential skill in understanding linear equations. To graph a linear equation such as \(y = 2x + 1\), we start by plotting the y-intercept on the y-axis. The y-intercept is the point where the line cuts through the y-axis, which, in this case, is \((0, 1)\). Once the y-intercept is plotted, we use the slope to determine the direction and steepness of the line. Since our slope here is '2', for every one unit you move right along the x-axis, you will move two units up along the y-axis. Plotting another point using this slope and drawing a line through these points will give you the graph of the equation.

Coefficient

The coefficient in a linear equation is the number that multiplies the variable \(x\). In the equation \(y = 2x + 1\), the coefficient is '2'. It's crucial because it directly affects the slope of the line when graphed, dictating whether the line slants upwards or downwards, and by how much. If the coefficient is positive, the line slopes upward as it moves from left to right. If negative, the line slopes downward. The larger the absolute value of the coefficient, the steeper the slope. So, a line with a coefficient of '5' would be steeper than one with a coefficient of '2'.

Y-axis Intercept

The y-axis intercept, also known just as the y-intercept, is the point where the graph of a line will cross the y-axis. This is represented by the 'b' in the linear equation formula \(y=mx+b\). This intercept provides not only a starting point to graph a linear equation but also gives us an idea of where the line will be positioned vertically at the point where \(x=0\). For the equation \(y=2x+1\), our y-intercept is '1', indicating that the line will cross the y-axis at the point \((0, 1)\), which can be seen as our starting point for plotting the graph of the equation.