Find the Equation Using Slope-Intercept Form point: (9,3) ; slope: 4/9

Solution:

Determine the Equation in Slope-Intercept Form

Given: Point (9,3), Slope = $\frac{4}{9}$

Step 1: Identify the y-intercept ($b$) using the line equation formula.

Step 1.1: Apply the slope-intercept form $y=mx+b$.

Step 1.2: Insert the slope ($m$) into the equation: $y=\left(\frac{4}{9}\right)x+b$.

Step 1.3: Plug in the $x$-coordinate from the given point: $y=\left(\frac{4}{9}\right)(9)+b$.

Step 1.4: Insert the $y$-coordinate to find $b$: $3=\left(\frac{4}{9}\right)(9)+b$.

Step 1.5: Solve for $b$.

Step 1.5.1: Rearrange the equation: $\frac{4}{9}\cdot 9+b=3$.

Step 1.5.2: Simplify by eliminating the common factor.

Step 1.5.2.1: Remove the common factor: $\frac{4}{\cancel{9}}\cdot \cancel{9}+b=3$.

Step 1.5.2.2: Simplify further: $4+b=3$.

Step 1.5.3: Isolate $b$ by moving other terms to the opposite side.

Step 1.5.3.1: Subtract $4$ from both sides: $b=3-4$.

Step 1.5.3.2: Calculate the difference: $b=-1$.

Step 2: With $m$ and $b$ now known, insert them back into $y=mx+b$ to get the line's equation: $y=\frac{4}{9}x-1$.

Knowledge Notes:

The slope-intercept form of a linear equation is one of the most common ways to represent a line. It is expressed as $y=mx+b$, where $m$ is the slope of the line and $b$ is the y-intercept, which is the point where the line crosses the y-axis.

• Slope ($m$): This is a measure of the steepness of the line. It is calculated as the rise over the run between two points on the line. In this case, the slope is given as $\frac{4}{9}$.

• Y-intercept ($b$): This is the value of $y$ when $x=0$. It can be found by substituting the $x$ and $y$ values of a point on the line (in this case, (9,3)) into the equation and solving for $b$.

• Substitution: To find $b$, we substitute the known values of $x$, $y$, and $m$ into the slope-intercept form and solve for $b$.

• Simplification: When simplifying the equation, we look for common factors that can be canceled out to make the equation easier to solve.

• Isolation of Variable: To solve for $b$, we isolate it on one side of the equation by performing inverse operations (like subtraction or addition) on both sides of the equation.

Once $b$ is found, we can write the final equation of the line using the slope-intercept form with our calculated values of $m$ and $b$.