Question
Find the equation of a line that passes through
and ![left parenthesis 2 comma negative 14 right parenthesis](data:image/png;base64,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)
Hint:
We are given two points and we need to find the equation of the line passing through them. The equation of a line passing through two points (a, b) and (c, d) is
The correct answer is: 3x + y + 8 = 0
![fraction numerator y minus d over denominator d minus b end fraction equals fraction numerator x minus c over denominator c minus a end fraction](data:image/png;base64,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)
Step by step solution:
Let the given points be denoted by
(a, b) = (-3, 1)
(c, d) = (2, -14)
The equation of a line passing through two points (a, b) and (c, d) is
![fraction numerator y minus d over denominator d minus b end fraction equals fraction numerator x minus c over denominator c minus a end fraction](data:image/png;base64,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)
Using the above points, we have
![fraction numerator y minus left parenthesis negative 14 right parenthesis over denominator negative 14 minus 1 end fraction equals fraction numerator x minus 2 over denominator 2 minus left parenthesis negative 3 right parenthesis end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKoAAAAnCAYAAACfWNOEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAZpE86XgAAA69JREFUeNrtnN+HVGEYx1/JSLrJyl50sayVJGPpaq21IslaY0WykqxlJSv9A+kiMbpYXXST7EXWiqwkWUuSjKwlSZJE+h/2Imss2/PYZ3LmdObMObPv87zPmfN8+DJz5jXve955zvv7O87p4Bao7nRSp/IZJacK2lRexo9UTqPEYBCMMn33KdBd0JeM6adBewnXz4Ea9lMlMg5aA22DmlTX1/rtJkcpULlYAS10CL44x0DfU9I2KGCNdj6AZqn+kDP0m872000+BC0K5JMlUJ+A5lPS3lY8jtbGEOirREbLoKsJ1++AHnjMZwM0oSBQsft61yXtWCSNduY7PFT36TMJdiQyuUGtXZSjoJ+gAY/54LimEjhQK/T0D3VJW6HyFoXPoMHI+znQY6G8x5iHdP+YAr2IXbtHExOf7ApVXFqg1mPDj7S0zQIF6iXQEr2+INgbHAFtUS/FDraa3yLvT4B+USE6BUI39RKovX5v1kCtJjz5/RKoyFvQJM3EBwTq+jjoFeii5E3+ibxeovGpb0J3/fjkj2RMW7SuvzWnaDqZNeBhCtIR6ZtsUObD1JoeYnrixwMGap4WpEiTqdYEcY0aGe6lotOgpzSPEecZqAZaBV1nykPT8lS3tIsJE0ytYOPymgIH1zc/eZ4ERxmk+czhkN3GsuNdD+Ne8PcZqEVZ8Mf5xHosMHHHbYUpvzfUogZjhn60GnM+m4xjqF4mBEXeQsVx9EvQyYTPntNKgEQd7+VsHA5EzckcFrFDKcaBWKcJhAQ3nd7tSRyXLlg46OQsjT0Mw+hjfG1UGIbRz62Aib9VtHq1nse6fsMwDMMwjHZCWNfNkm7kIuQuoardP1/25ZBlkkTaisxhXT/v9nc58Xwz+qjQzvTI/X+aS9V5Cp/25RBlkkbSisx1ku096LJrPyCPeW0kpFV3Qs2HfTlEmTTAZUWWOhvcIsktoc6S7sO+XNZAdY7HiixlXUfwYPePhOvqXBQ+7MtlDVQuK7KEfw3dAHM0Tp3q8NtvFyVQ89iXyxaonFbkXea6jSrNKNrkzDjvFqEv+7KPshQlULNakTVb1/EeZuhhm5QIVK7WK499uUwtqoQVWcq67mgVY6vIXX/IgxxaA1XKiixlXU+bEBZqMhUqgDQGqqQVWXJ5qkoTqjjqLOkWqNmQtCJzLfjjpsUVetjwT0xwp+q32/8jvjhqFvx92ZdDlylkfXGWkcO6PkEP3A4Jd6qmE9LZv3obubpkO5RiFIIQ1vWulvS/aFzvHZ1Q1BAAAAGMdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtcm93PjxtaT55PC9taT48bW8+JiN4MjIxMjs8L21vPjxtbz4oPC9tbz48bW8+JiN4MjIxMjs8L21vPjxtbj4xNDwvbW4+PG1vPik8L21vPjwvbXJvdz48bXJvdz48bW8+JiN4MjIxMjs8L21vPjxtbj4xNDwvbW4+PG1vPiYjeDIyMTI7PC9tbz48bW4+MTwvbW4+PC9tcm93PjwvbWZyYWM+PG1vPj08L21vPjxtZnJhYz48bXJvdz48bWk+eDwvbWk+PG1vPiYjeDIyMTI7PC9tbz48bW4+MjwvbW4+PC9tcm93Pjxtcm93Pjxtbj4yPC9tbj48bW8+JiN4MjIxMjs8L21vPjxtbz4oPC9tbz48bW8+JiN4MjIxMjs8L21vPjxtbj4zPC9tbj48bW8+KTwvbW8+PC9tcm93PjwvbWZyYWM+PC9tYXRoPkZeM7IAAAAASUVORK5CYII=)
Simplifying the above equation, we have
![fraction numerator y plus 14 over denominator negative 15 end fraction equals fraction numerator x minus 2 over denominator 2 plus 3 end fraction](data:image/png;base64,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)
![not stretchy rightwards double arrow fraction numerator y plus 14 over denominator negative 15 end fraction equals fraction numerator x minus 2 over denominator 5 end fraction](data:image/png;base64,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)
Cross multiplying, we get
5(y + 14) = -15(x - 2)
Expanding the factors, we have
5y + 70 = -15x + 30
Taking all the terms in the left hand side, we have
15x + 5y + 70 - 30 = 0
Finally, the equation of the line is
15x + 5y + 40 = 0
Dividing the equation throughout by 5, we get
3x + y + 8 = 0
This is the required equation.
Step by step solution:
Let the given points be denoted by
The equation of a line passing through two points (a, b) and (c, d) is
Using the above points, we have
Simplifying the above equation, we have
Cross multiplying, we get
Expanding the factors, we have
Taking all the terms in the left hand side, we have
Finally, the equation of the line is
Dividing the equation throughout by 5, we get
This is the required equation.
We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by ax + by + c = 0,, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.