Problem 1 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((1,5)\) with slope 3

Step-by-Step Solution

Answer

The equation of the line is \( y = 3x + 2 \).

Step 1 - Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

Step 2 - Substitute the Given Slope

Given the slope \( m = 3 \), the equation becomes: \[ y = 3x + b \]

Step 3 - Substitute the Point Coordinates

Use the point \( (1, 5) \). Substitute \( x = 1 \) and \( y = 5 \) into the equation: \[ 5 = 3(1) + b \]

Step 4 - Solve for the Y-Intercept

Solve the equation \[ 5 = 3 + b \] to find the value of \( b \). Subtract 3 from both sides: \[ 5 - 3 = b \] Thus, \( b = 2 \).

Step 5 - Write the Final Equation

Substitute \( b = 2 \) back into the slope-intercept form: \[ y = 3x + 2 \]

Key Concepts

slope-intercept form

To understand how to write an equation of a line, you must first know about the slope-intercept form. This is a way to describe straight lines using an easy-to-remember equation:

Definition

The slope-intercept form of a line is given by the equation: $y\; =\; mx\; +\; b$
Here, $m$ represents the slope of the line and $b$ represents the y-intercept.

Slope

The slope, $m$, shows how steep the line is. If the slope is positive, the line goes upwards. If it's negative, the line goes downwards.

Y-Intercept

The y-intercept, $b$, is the point where the line crosses the y-axis. This happens when $x\; =\; 0$. By understanding these parts, we can start to construct our equation.

substitute coordinates

Once we know our slope, we need to fit it into our equation. Let’s use an example point and incorporate it.

Given Slope

In our exercise, the slope is given as $3$. Start with the slope-intercept form: $y\; =\; mx\; +\; b$.
Substitute $m\; =\; 3$: $y\; =\; 3x\; +\; b$

Example Point

Next, we use the given point, which is $(1,5)$. The coordinates tell us that when $x\; =\; 1$, $y\; =\; 5$. Put these values into the equation: $5\; =\; 3(1)\; +\; b.$

Why Substitute?

Substituting these coordinates into the equation helps us to find the missing value, which is the y-intercept. This way, each part of the equation falls into place.

solve for y-intercept

Now that we have substituted our values, we can solve for the y-intercept. This will give us the full equation of the line.

Simplify the Equation

After substitution, we have this: $5\; =\; 3(1)\; +\; b$
Simplify the right side: $5\; =\; 3\; +\; b.$

Solving Steps

To isolate $b$, subtract 3 from both sides: $5\; -\; 3\; =\; b.$
This results in $b\; =\; 2.$ Finally, substitute $b\; =\; 2$ into the equation:
Thus, the full equation is $y\; =\; 3x\; +\; 2.$

Review

By solving for the y-intercept, we completed our equation. We now have our final answer, that describes the line passing through $(1,\; 5)$ with a slope of $3.$ Understanding this process makes it easier to handle different line equations in future exercises.