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Linear equations in two variables SAT question

Math · Algebra · Hard · Answer it here, explanation included.

In the xy-plane, line k intersects the y-axis at the point $with coordinates 0 comma negative 6$ and passes through the point $with coordinates 2 comma 2$ . If the point $with coordinates 20 comma w$ lies on line k, what is the value of w ?

Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.

ExplanationShow

The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair $x comma y$ of the point of intersection of the line with the y-axis. Since line k intersects the y-axis at the point $with coordinates 0 comma negative 6$ , it follows that $the point with coordinates 0 comma negative 6$ is the y-intercept of this line. An equation of any line in the xy-plane can be written in the form $y equals, m x plus b$ , where m is the slope of the line and b is the y-coordinate of the y-intercept. Therefore, the equation of line k can be written as $y equals, m x plus negative 6$ , or $y equals, m x minus 6$ . The value of m can be found by substituting the x- and y-coordinates from a point on the line, such as $the point with coordinates 2 comma 2$ , for x and y, respectively. This results in $2 equals, 2 m minus 6$ . Solving this equation for m gives $m equals 4$ . Therefore, an equation of line k is $y equals, 4 x minus 6$ . The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for y. This gives $y equals, 4 times 20, minus 6$ , or $y equals 74$ . Since w is the y-coordinate of this point, $w equals 74$ .

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Linear equations in two variables SAT question

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