Question

If the slope of the line shown is 3 , then what is the \(x\) coordinate of point \(\mathrm{B}\) ? A. -5 B. -3 C. 3 D. 5

Short answer

The x-coordinate of point B is -3.

Step-by-step solution

Identify the given information

We are given the slope of the line = 3. Also, we know that one point on the line is A(-3, 1) and we need to find the x-coordinate of point B. Let's assume the coordinates of point B are (x, y).

Use point slope form of the equation to find the x-coordinate

The point slope form of the equation of a line is given by: (y - y1) = m(x - x1), where m is the slope and (x1, y1) are the coordinates of point A. We have m = 3, and point A has coordinates (-3, 1). Let's substitute the value of m and the coordinates of point A in the point slope equation to find the equation of the line. (y - 1) = 3(x - (-3)) Now we need to find a point on the line, using the possible x-coordinates provided in the options (A to D).

Check each option

Let's substitute each potential x-coordinate from the options into our equation and see which one gives us an integer value for y. A) x = -5: (y - 1) = 3(-5 - (-3)) => (y - 1) = 3(-2) => y = -5 B) x = -3: (y - 1) = 3(-3 - (-3)) => (y - 1) = 3(0) => y = 1 C) x = 3: (y - 1) = 3(3 - (-3)) => (y - 1) = 3(6) => y = 19 D) x = 5: (y - 1) = 3(5 - (-3)) => (y - 1) = 3(8) => y = 25

Determine the correct x-coordinate

Among the options A to D, only option B gives us an integer value for y when we substitute the x-coordinate into the equation. Thus, the x-coordinate of point B is -3. The correct answer is B) -3.

Key concepts

Point-Slope Form

The point-slope form of a line is a powerful tool in coordinate geometry. It helps you find the equation of a line when you know the slope and at least one point on the line. The formula is: \( (y - y_1) = m(x - x_1) \) Here,

• \(y - y_1\) is the difference in the y-coordinates
• \(m\) is the slope of the line
• \(x - x_1\) is the difference in the x-coordinates

You can use this form to derive linear equations quickly. For instance, if you know a line passes through point \((-3, 1)\) with a slope of 3, you replace these values into the formula: \((y - 1) = 3(x + 3)\). Such versatility makes point-slope form an essential concept for solving problems involving linear equations with limited data.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through a coordinate system. Typically, this involves plotting points on the Cartesian plane, which consists of the x-axis (horizontal) and y-axis (vertical). In coordinate geometry, understanding the relationship between points, lines, and shapes becomes easier as we can use algebraic methods to solve geometric problems.
Using the coordinates of points, we can calculate distances, find midpoints, and determine slopes. For example, knowing the coordinates of a point directly helps in finding the equation of a line using the point-slope form. Imagine knowing one point on a line, you can use its coordinates to find or predict another point given the line's slope. This makes it possible not just to verify calculations, but also enables practical applications in computer graphics and various fields of engineering.

Linear Equations

Linear equations represent straight lines in a graph. These equations typically have variables with a power of one and follow a form like \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. Linear equations are foundational in mathematics because they often model real-world situations and relationships, such as speed, cost, or distance over time. They are used in various fields from economics to physics.
In coordinate geometry, understanding linear equations helps you predict values, like finding unknown coordinates when given a slope and one point on a line. For instance, rearranging the point-slope formula or setting up the standard form can provide the linear equation of a line. Such equations make it easy to predict how a line extends infinitely in both directions and helps in solving systems of equations that use multiple lines and points.