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Nonlinear equations in one variable and systems of equations in two variables SAT question

Math · Advanced Math · Medium · Answer it here, explanation included.

In the xy-plane, what is the y-coordinate of the point of intersection of the graphs of $y equals, open parenthesis, x minus 1, close parenthesis, squared$ and $y equals, 2 x minus 3$ ?

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

ExplanationShow

The correct answer is 1. The point of intersection of the graphs of the given equations is the solution to the system of the two equations. Since $y equals, open parenthesis, x minus 1, close parenthesis, squared$ and $y equals, 2 x minus 3$ , it follows that $open parenthesis, x minus 1, close parenthesis, squared, equals, 2 x minus 3$ , or $open parenthesis, x minus 1, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, equals, 2 x minus 3$ . Applying the distributive property to the left-hand side of this equation yields $x squared, minus 2 x, plus 1, equals, 2 x minus 3$ . Subtracting $2 x$ from and adding 3 to both sides of this equation yields $x squared, minus 4 x, plus 4, equals 0$ . Factoring the left-hand side of this equation yields $open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 2, close parenthesis, equals 0$ . By the zero product property, if $open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x minus 2, close parenthesis, equals 0$ , it follows that $x minus 2, equals 0$ . Adding 2 to both sides of $x minus 2, equals 0$ yields $x equals 2$ . Substituting 2 for x in either of the given equations yields $y equals 1$ . For example, substituting 2 for x in the second given equation yields $y equals, 2 times 2, minus 3$ , or $y equals 1$ . Therefore, the point of intersection of the graphs of the given equations is $the point with coordinates 2 comma 1$ . The y-coordinate of this point is 1.

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Nonlinear equations in one variable and systems of equations in two variables SAT question

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