Question number 17. How many pairs of real numbers x y satisfy the following equation? Let's begin by expanding both sides of the equation and then we'll decide what to do next. On the left hand side we have x squared plus 2xy plus y squared. On the right hand side we obtain x times y minus 5x plus 5y minus 25 and then we can group the terms together on one side. Now we have 1xy on the left hand side and on the right hand side the minus 25 and we might try to complete the square here on the x terms that would be possible but on the y terms we run into trouble with the term xy so this suggests a change of variable. Our equation resembles the equation of the conic section but with a xy term there is some sort of a rotation so we will let x plus 5 be a new variable we'll call it u we'll let y minus 5 be another variable we'll call it v and so by adding we see that u plus v is equal to x plus y and then we have our new equation instead of the original one an equivalent equation in the new variables u and v would be u plus v quantity squared is equal to u times v and then we can work with this equation expanding the left hand side we have a u squared plus 2uv plus v squared on the right hand side the u times v which we then subtract and our simplified equation in u and v is u squared plus uv plus v squared is equal to 0 we note that if both u and v are not equal to 0 then this equation will have no solutions each term will be nonzero so we have to suppose that u and v are 0 one at a time so we let be equal to 0 then the equation becomes v squared is equal to 0 so v has to be 0 alternatively if we let v be equal to 0 then the equation becomes u squared is equal to 0 and so u is also equal to 0 so we have no choice there is just one solution and that's when u is equal to 0 and v is equal to 0 so only the point 0 comma 0 or the origin belongs to the graph of this equation and the answer is B

real number pairs

conic form

change of variable

equation simplification

unique solution