Let's understand the linear equation y equals 2x plus 1. This equation represents a straight line on a coordinate plane. In the form y equals mx plus b, the number 2 is the slope, which tells us that for every 1 unit increase in x, y increases by 2 units. The number 1 is the y-intercept, which is the value of y when x equals zero. This is the point where the line crosses the y-axis at zero comma one. For example, when x equals 2, y equals 2 times 2 plus 1, which equals 5. So the point 2 comma 5 lies on our line.
Now let's understand what the slope of 2 really means. The slope measures the steepness of a line and is calculated as rise divided by run. Rise is the vertical change, or change in y, while run is the horizontal change, or change in x. Looking at our line y equals 2x plus 1, let's pick two points: the point at x equals 1, where y equals 3, and the point at x equals 3, where y equals 7. The rise between these points is 4 units up, and the run is 2 units to the right. Dividing rise by run gives us 4 divided by 2, which equals 2. This confirms that our line has a slope of 2, meaning that y increases by 2 units for every 1 unit increase in x.
Now let's focus on the y-intercept of our linear equation y equals 2x plus 1. The y-intercept is the point where the line crosses the y-axis, which occurs when x equals zero. In our equation y equals 2x plus 1, when we substitute x equals zero, we get y equals 2 times 0 plus 1, which equals 1. So the y-intercept is at the point zero comma one. The y-intercept represents the initial value or starting point of the relationship. As we move along the line, for any input value of x, we can find the corresponding output value y by substituting into our equation. For example, when x equals negative 3, y equals 2 times negative 3 plus 1, which equals negative 5. As x increases, y increases at a rate determined by the slope, which is 2 in our equation.
Let's explore a real-world application of our linear equation y equals 2x plus 1. Imagine a cyclist traveling along a straight path. In this scenario, y represents the distance in miles from a reference point, and x represents the time in hours. The equation y equals 2x plus 1 tells us that the cyclist travels at a constant speed of 2 miles per hour, which is the slope of our line. The y-intercept of 1 means that at time zero, the cyclist was already 1 mile away from the reference point. As time passes, we can track the cyclist's position. After 1 hour, they've traveled to the 3-mile mark. After 2 hours, they're at the 5-mile mark. After 3 hours, they've reached the 7-mile mark, and after 4 hours, they're at the 9-mile mark. This demonstrates how linear equations can model constant-rate scenarios like uniform motion.
Let's summarize what we've learned about the linear equation y equals 2x plus 1. This equation represents a straight line on a coordinate plane. The coefficient 2 is the slope, which tells us that for every unit increase in x, y increases by 2 units. The constant term 1 is the y-intercept, which means the line crosses the y-axis at the point zero comma one. Linear equations like this one are powerful tools for modeling real-world scenarios with constant rates of change, such as uniform motion, simple interest, or direct proportional relationships. Once we understand the equation, we can use it to make predictions by calculating the y-value for any given x-value. This fundamental relationship between variables forms the basis for more complex mathematical models in science, economics, engineering, and many other fields.