Find an equation for the line with the given property.
(a) It passes through the point $(5,-7)$, and is parallel to the line $2 x+y-10=0$
(b) It has $x$-intercept 6 and $y$-intercept 4.
Answer (1)
For part (a), find the slope of the parallel line 2 x + y − 10 = 0 , which is − 2 . Use the point-slope form with the point ( 5 , − 7 ) to find the equation: y = − 2 x + 3 .
For part (b), calculate the slope using the x-intercept ( 6 , 0 ) and y-intercept ( 0 , 4 ) : m = − 3 2 . Use the slope-intercept form to find the equation: y = − 3 2 x + 4 .
The equation of the line in part (a) is y = − 2 x + 3 .
The equation of the line in part (b) is y = − 3 2 x + 4 .
Explanation
Problem Overview We are given two separate problems, each asking for the equation of a line based on different criteria. We will solve each part independently.
Finding the Slope (a) We need to find the equation of a line that passes through the point ( 5 , − 7 ) and is parallel to the line 2 x + y − 10 = 0 . First, let's find the slope of the given line. We can rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope.
Parallel Line Slope The given line is 2 x + y − 10 = 0 . Subtracting 2 x and adding 10 to both sides, we get y = − 2 x + 10 . Thus, the slope of the given line is − 2 . Since the line we want to find is parallel to this line, it will have the same slope, which is m = − 2 .
Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point ( 5 , − 7 ) and m is the slope − 2 . Substituting these values, we get y − ( − 7 ) = − 2 ( x − 5 ) .
Equation of Line (a) Simplifying the equation, we have y + 7 = − 2 x + 10 . Subtracting 7 from both sides, we get y = − 2 x + 3 .
Intercepts (b) We need to find the equation of a line with x -intercept 6 and y -intercept 4. The x -intercept is the point where the line crosses the x -axis, which is ( 6 , 0 ) . The y -intercept is the point where the line crosses the y -axis, which is ( 0 , 4 ) .
Calculating Slope We can find the slope of the line passing through these two points using the formula m = x 2 − x 1 y 2 − y 1 . Substituting the coordinates of the points ( 6 , 0 ) and ( 0 , 4 ) , we get m = 0 − 6 4 − 0 = − 6 4 = − 3 2 .
Equation of Line (b) Now we use the slope-intercept form of a line, which is y = m x + b , where m is the slope and b is the y -intercept. We already know the y -intercept is 4, so b = 4 . Substituting the slope m = − 3 2 and the y -intercept b = 4 , we get y = − 3 2 x + 4 .
Final Answer Therefore, the equation of the line in part (a) is y = − 2 x + 3 , and the equation of the line in part (b) is y = − 3 2 x + 4 .
Examples
Understanding linear equations is crucial in many real-world applications. For example, in business, a linear equation can represent the relationship between the number of products sold and the profit earned. Similarly, in physics, it can describe the motion of an object moving at a constant velocity. By finding the equation of a line given certain conditions, we can model and predict various phenomena in these fields, allowing for informed decision-making and problem-solving.
