Find an equation for the line with the given information:
(a) It passes through the point (5, -7) and is parallel to the line [tex]2x + y - 10 = 0[/tex].
(b) It has x-intercept 6 and y-intercept 4.
Answer (1)
For part (a), find the slope of the given line and use the point-slope form to find the equation of the parallel line: y = − 2 x + 3 .
For part (b), calculate the slope using the x and y intercepts and use the slope-intercept form to find the equation of the line: 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 Analysis We are given two separate problems. In the first, we need to find the equation of a line that passes through a given point and is parallel to another line. In the second, we need to find the equation of a line given its x and y intercepts.
Solving Part (a) For part (a), we first need to find the slope of the line 2 x + y − 10 = 0 . We can rewrite this equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. So, y = − 2 x + 10 . The slope of this line is -2. Since parallel lines have the same slope, the line we are looking for also has a slope of -2. We know that the line passes through the point ( 5 , − 7 ) . We can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. Plugging in the values, we get y − ( − 7 ) = − 2 ( x − 5 ) , which simplifies to y + 7 = − 2 x + 10 . Subtracting 7 from both sides, we get y = − 2 x + 3 .
Solving Part (b) For part (b), we are given the x-intercept as 6 and the y-intercept as 4. This means the line passes through the points ( 6 , 0 ) and ( 0 , 4 ) . We can find the slope of the line using the formula m = x 2 − x 1 y 2 − y 1 . Plugging in the values, we get m = 0 − 6 4 − 0 = − 6 4 = − 3 2 . Since we know the y-intercept is 4, we can 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. Plugging in the values, 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 instance, in business, you might use a linear equation to model the cost of producing a certain number of items, where the slope represents the variable cost per item, and the y-intercept represents the fixed costs. Similarly, in physics, you can use linear equations to describe the motion of an object moving at a constant velocity, where the slope represents the velocity, and the y-intercept represents the initial position. These models help in making predictions and informed decisions.
