Question

What are the intercepts of \(y=5 x+10 ?\)

Short answer

The intercepts are (0, 10) and (-2, 0).

Step-by-step solution

Identify the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. Solve for y.

Calculate the y-intercept

Substitute x = 0 into the equation: \[ y = 5(0) + 10 \] This simplifies to \[ y = 10 \] Thus, the y-intercept is (0, 10).

Identify the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. Solve for x.

Calculate the x-intercept

Substitute y = 0 into the equation: \[ 0 = 5x + 10 \] Solve for x: \[ -10 = 5x \] \[ x = -2 \] Thus, the x-intercept is (-2, 0).

Key concepts

y-intercept

The y-intercept is the point where a line or curve crosses the y-axis on a graph. To find the y-intercept of a linear equation, you need to set the value of x to 0 and solve for y. When x is zero, any term involving x will disappear, simplifying your calculations. For example, in the equation provided, you substitute x with 0:

1. Substitute x = 0:

    y = 5(0) + 10 = 10 

2. Therefore, the y-intercept is (0, 10) because when x is 0, y is 10.

Understanding intercepts helps in graphing and interpreting functions. It's worth noting that the y-intercept signifies the starting position of a line when graphed.

x-intercept

The x-intercept is the point where a line or curve crosses the x-axis on a graph. To find the x-intercept of a linear equation, set the value of y to 0 and solve for x. When y is zero, you eliminate the y term, which simplifies the equation. In the provided example, you find the x-intercept as follows:

1. Substitute y = 0:

    0 = 5x + 10 

2. Solve for x:

    -10 = 5x  x = -2 

This indicates that the x-intercept is (-2, 0) because when y is 0, x is -2. Skipping this step might result in errors in graphing or interpreting functions.

linear equation

A linear equation is an equation that makes a straight line when it is graphed. These equations can be written in the standard form y = mx + b, where m is the slope and b is the y-intercept. Here are some essential points to keep in mind:

• Linear equations describe a constant rate of change.
• They are characterized by their straight-line graphs.
• The slope (m) represents the steepness of the line and direction, either positive or negative.

In the given example, the equation y = 5x + 10 has a slope of 5 and a y-intercept of 10, meaning the line rises 5 units up for each unit it moves right. Understanding linear equations is crucial for solving real-world problems involving constant rates and linear relationships.