Question

Describe the equations of the linear functions that go through the point \((1,3) .\) Give 2 examples.

Short answer

Examples: y = 2x + 1 and y = -x + 4.

Step-by-step solution

Understanding the Problem

We need to find equations of two linear functions that pass through the point (1,3). A linear function is usually expressed in the form of y = mx + b, where m is the slope and b is the y-intercept.

Choosing Slopes for Example 1

For the first example, let's choose a slope of m = 2. We substitute the point (1,3) into the line equation y = 2x + b to find b.

Solving for the Y-Intercept (b) in Example 1

Substitute (1,3) into the equation: 3 = 2(1) + b Solve for b: 3 = 2 + b b = 1. So, the equation for the first line is y = 2x + 1.

Choosing Slopes for Example 2

For the second example, let's choose a different slope of m = -1. We use the same approach to find b using the point (1,3).

Solving for the Y-Intercept (b) in Example 2

Substitute (1,3) into the equation: 3 = -1(1) + b Solve for b: 3 = -1 + b b = 4. So, the equation for the second line is y = -x + 4.

Key concepts

Slope-Intercept Form

Linear equations are often expressed in the slope-intercept form, which is written as \( y = mx + b \). This form shows how the line behaves on a graph.
Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept.

• The slope \( m \) tells us how steep the line is and in which direction it tilts.
• The y-intercept \( b \) is where the line crosses the y-axis.

Understanding the slope-intercept form helps us easily plot linear equations and predict the behavior of the line just by looking at the values of \( m \) and \( b \).For example, consider an equation like \( y = 2x + 1 \). Here, the slope \( m = 2 \) indicates the line rises vertically two units for every unit it moves horizontally. The y-intercept \( b = 1 \) tells us the line crosses the y-axis at \( y=1 \).

Point-Slope Relationship

The point-slope form is a useful tool for finding the equation of a line when you know a point on the line and the slope. It is expressed as \( y - y_1 = m(x - x_1) \).

• \( (x_1, y_1) \) is a known point on the line.
• \( m \) is the slope.

This form helps in deriving a line equation without initially knowing the y-intercept.For instance, given a point like \((1,3)\) and a slope \( m = 2 \), you can plug these values into the point-slope form:\[ y - 3 = 2(x - 1)\]When you simplify it, you'll obtain the slope-intercept form \( y = 2x + 1 \).
Notice how this connection allows us to flexibly adapt and use given information to create the line equation.

Y-Intercept Calculation

The y-intercept in a linear equation is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) is the value of the y-intercept.It is crucial for plotting the line on a graph and understanding its position.
You can calculate \( b \) using the known values of a point \((x, y)\) and the slope \( m \).
Here's how:

• Start with the equation \( y = mx + b \).
• Substitute the values into the equation, i.e., \( y = mx + b \).
• Solve for \( b \). This means rearranging the equation to get \( b \) alone on one side.

Consider a point \((1, 3)\) on a line with a slope \( m = -1 \):\[3 = -1 \, (1) + b \] Solve for \( b \):\[3 = -1 + b \ \rightarrow b = 4\]So, the line crossing this point with \( m = -1 \) has a y-intercept at \( y = 4 \), giving us the equation \( y = -x + 4 \).
Mastering how to find \( b \) is essential for forming and interpreting linear equations.