Write the equation of a line that passes through $(-6,2)$ and is parallel to $\frac{3}{2} x+8$. Express your answer in slope-intercept form.
Determine if the lines $y=2 x+5$ and $4 x-2 y=10$ are parallel, perpendicular, or the same line.
Answer (2)
Find the slope of the parallel line: Since the line is parallel to y = 2 3 x + 8 , the slope is 2 3 .
Use the point-slope form: Substitute the point ( − 6 , 2 ) into the equation y = 2 3 x + b .
Solve for the y-intercept: 2 = 2 3 ( − 6 ) + b ⇒ b = 11 .
Write the equation in slope-intercept form: y = 2 3 x + 11 . The lines y = 2 x + 5 and 4 x − 2 y = 10 are parallel. y = 2 3 x + 11 and Parallel.
Explanation
Understanding the Problem We are given a point ( − 6 , 2 ) and a line y = 2 3 x + 8 . We need to find the equation of a line that passes through the given point and is parallel to the given line. The equation must be in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope Since the line we want to find is parallel to y = 2 3 x + 8 , it has the same slope. Therefore, the slope of our line is m = 2 3 . So, the equation of our line is y = 2 3 x + b .
Substituting the Point Now we need to find the y-intercept, b . We know that the line passes through the point ( − 6 , 2 ) . We can substitute these values into the equation y = 2 3 x + b to solve for b :
2 = 2 3 ( − 6 ) + b
Simplifying the Equation Simplify the equation: 2 = − 9 + b
Solving for b Add 9 to both sides to solve for b :
2 + 9 = b b = 11
Writing the Equation Now we have the slope m = 2 3 and the y-intercept b = 11 . We can write the equation of the line in slope-intercept form: y = 2 3 x + 11
Analyzing the Second Pair of Lines Now, let's determine if the lines y = 2 x + 5 and 4 x − 2 y = 10 are parallel, perpendicular, or neither. First, we need to rewrite the second equation in slope-intercept form: 4 x − 2 y = 10 Subtract 4 x from both sides: − 2 y = − 4 x + 10 Divide both sides by − 2 :
y = 2 x − 5
Determining the Relationship The first line is y = 2 x + 5 , which has a slope of 2 . The second line is y = 2 x − 5 , which also has a slope of 2 . Since the slopes are equal, the lines are parallel.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, in economics, you might use a linear equation to model the relationship between the price of a product and the quantity demanded. If you know the slope (rate of change) and a specific point (price and quantity), you can determine the entire demand curve. Similarly, in physics, linear equations can describe motion with constant velocity, where the slope represents the velocity and the y-intercept represents the initial position. These models help predict and analyze various phenomena in a simplified yet effective manner.
The equation of the line passing through ( − 6 , 2 ) and parallel to y = 2 3 x + 8 is y = 2 3 x + 11 . The lines y = 2 x + 5 and 4 x − 2 y = 10 are parallel.
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