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Algebra / Linear equations in two variables Difficulty: Hard

The graph of $9 x - 10 y = 19$ is translated down $4$ units in the xy-plane. What is the x-coordinate of the x-intercept of the resulting graph?

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Explanation

The correct answer is $StartFraction 59 Over 9 EndFraction$ . When the graph of an equation in the form $A x + B y = C$ , where $A$ , $B$ , and $C$ are constants, is translated down $k$ units in the xy-plane, the resulting graph can be represented by the equation $A x + B (y + k) = C$ . It’s given that the graph of $9 x - 10 y = 19$ is translated down $4$ units in the xy-plane. Therefore, the resulting graph can be represented by the equation $9 x - 10 (y + 4) = 19$ , or $9 x - 10 y - 40 = 19$ . Adding $40$ to both sides of this equation yields $9 x - 10 y = 59$ . The x-coordinate of the x-intercept of the graph of an equation in the xy-plane is the value of $x$ in the equation when $y equals 0$ . Substituting $0$ for $y$ in the equation $9 x - 10 y = 59$ yields $9 x - 10 (0) = 59$ , or $9 x equals 59$ . Dividing both sides of this equation by $9$ yields $x = \frac{59}{9}$ . Therefore, the x-coordinate of the x-intercept of the resulting graph is $StartFraction 59 Over 9 EndFraction$ . Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.