Find the equation in slope intercept form for a line with [tex]$m =\frac{2}{3}$[/tex] and a point on the line of (-9,5).
A. [tex]$y=\frac{2}{3} x+11$[/tex]
B. [tex]$y=\frac{2}{3} x+1$[/tex]
C. [tex]$y=-9 x+11$[/tex]
D. [tex]$y=-9 x+1$[/tex]
Answer (2)
Substitute the given slope m = 3 2 into the slope-intercept form: y = 3 2 x + b .
Substitute the coordinates of the given point ( − 9 , 5 ) into the equation: 5 = 3 2 ( − 9 ) + b .
Solve for b : 5 = − 6 + b , so b = 11 .
Write the equation in slope-intercept form: y = 3 2 x + 11 . The final answer is y = 3 2 x + 11 .
Explanation
Understanding the Problem We are given the slope m = 3 2 and a point ( − 9 , 5 ) on the line. We want to find the equation of the line in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Substituting the Slope First, substitute the given slope m = 3 2 into the slope-intercept form: y = 3 2 x + b
Substituting the Point Next, substitute the coordinates of the given point ( − 9 , 5 ) into the equation: 5 = 3 2 ( − 9 ) + b
Solving for the y-intercept Now, solve for b : 5 = − 6 + b Add 6 to both sides: 5 + 6 = b 11 = b So, b = 11 .
Writing the Equation Finally, write the equation in slope-intercept form using the values we found for m and b : y = 3 2 x + 11
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the cost of a taxi ride, the initial fare is the y-intercept, and the cost per mile is the slope. By knowing these two values, you can predict the total cost of any ride. Similarly, in business, linear equations can model revenue and expenses, helping to forecast profits based on sales volume.
To find the equation of the line in slope-intercept form, we substitute the slope m = 3 2 and the point ( − 9 , 5 ) to find the y-intercept b = 11 . Therefore, the equation is y = 3 2 x + 11 . The correct option is A.
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