<p>Can someone just explain how to solve these?</p>
<p>1) A carpentry class made key-chain ornaments in the shape of squares,circles, and triangles to sell at a local fair. The class made a total of 2500 ornaments. IF the class made twice as many circles as squares and three times as many squares as triangles, how man trianges did the class make?</p>
<p>Ans: 250</p>
<p>-----A(20)------B-----X--C---------D(44)
2) The points A,B,C, and D(In that order) are equally spaced on the number line above, and X is anoother point on the line. If the distance from X to C is 2, what is the coordinates of the point (Not shown) that bisects line segment AX. </p>
<p>Ans: 27</p>
<p>3) If (x+a)(7x+b) = 7x^2 + cx + 6 for all values of x, and if a and b are positive integers, what is one possible value of c?</p>
<p>Ans: 5</p>
<p>1)</p>
<p>Let the number of triangles be X.
Three times as many squares as triangles were made. Let the number of circles be 3X.
Twice as many circles as squares were made. Let the number of triangles be 6X.</p>
<p>Their sum is 2500.</p>
<p>So 6X + 3X + X = 2500. The value of X is found, which is the value of the number of triangles. 250.</p>
<p>2) The distance between A and D is 44-20 = 24. AB, AC, and CD are all equal. That means each of them has a length of 8 each. Cut the length of BC by 2, since the distance between X and C is 2, so we have the length of BX which is 6. Therefore, the distance between A and X is 8 + 6 = 14. Now, what is the coordinate that would cut this line in half? It would be 20 + 7(since 7 is half of 14) = 27.</p>
<p>3) (x+a)(7x+b) = 7x^2 + cx + 6.
Time for algebra. Let’s hope you know your concepts and are able to follow.</p>
<p>7x^2 + bx + 7ax + ab = 7x^2 + cx + 6</p>
<p>We can deduce that ab = 6, and bx + 7ax = cx. Let’s put random values for a and b, in which the product will be 6. Let’s say 2 and 3? Alright. Now let’s plug them into the equation we actually want.</p>
<p>3x + 7(2)x = cx
3x + 14x = cx
17x = cx</p>
<p>Therefore c = 17 which is one of the possible solutions. The solutions vary according to the different values you may set for ab=6, IE (1,6), (2,3)</p>
<p>Yea, you know, I love questions in which an equation is true for all values of x. You can just plug in and have fun.</p>
<p>First off plug in for 0 and you get ab=6. This is only possible for 1<em>6, 6</em>1, 2<em>3 and 3</em>2.
Plug in x=1 and you get 7a+b=c. So when a is 2 and b is 3, c is 17.</p>