Find the Equation Using Slope-Intercept Form point: (9,3) ; slope: 4/9
The problem asks you to determine the equation of a straight line in slope-intercept form, which is y = mx + b, where 'm' represents the slope of the line and 'b' is the y-intercept. You are provided with two critical pieces of information to construct this equation: a point through which the line passes, given as coordinates (9, 3), and the slope of the line, which is 4/9. Using these, you're expected to plug in the values and manipulate the equation to find 'b', the y-intercept, thereby completing the equation.
point:$\left(\right. 9 , 3 \left.\right)$; slope:$\frac{4}{9}$
Answer
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 $b$.
