Forming and Classifying Pairs: Practice

How to use this set#

Here are ten ques­tions on turn­ing every­day sit­u­a­tions into a pair of lin­ear equa­tions, on the three cases, and on the coef­fi­cient test. A few ask you to draw, but most ask you to clas­sify a pair or find a miss­ing con­stant. One habit to keep through­out: put each equa­tion into the form $ax + by + c = 0$ before you read off its coef­fi­cients.

Sec­tion A, Form­ing the equa­tions#

1. A pen and two note­books cost ₹ 70. Three pens and one note­book cost ₹ 60. Write a pair of lin­ear equa­tions, tak­ing a pen to cost ₹ $x$ and a note­book ₹ $y$.
2. Meera is $6$ years older than her brother. In $4$ years, the sum of their ages will be $34$. Write the pair of equa­tions for their present ages.
3. The perime­ter of a rec­tan­gle is $28$ cm and its length is $2$ cm more than its width. Form the equa­tions.

Sec­tion B, Clas­si­fy­ing#

4. Say whether each pair is con­sis­tent, incon­sis­tent, or depen­dent: (a) $x - 4y + 2 = 0$ and $3x - 12y + 6 = 0$; (b) $2x + 5y = 7$ and $4x + 10y = 9$; (c) $x + y = 3$ and $x - y = 1$.
5. Draw the lines $2x + y = 6$ and $x + y = 4$ and read off their point of inter­sec­tion. Then con­firm it alge­braically.
6. Draw $x - y = 2$ and $2x - 2y = 8$. Describe what you see, and state the num­ber of solu­tions.

Sec­tion C, Find­ing a con­stant#

7. Find $p$ so that $2x + 3y = 7$ and $px + 6y = 14$ have infi­nitely many solu­tions.
8. Find $m$ so that $mx + 3y = 4$ and $2x + 6y = 9$ have no solu­tion.
9. Find all val­ues of $k$ for which $kx + 4y = 1$ and $3x + 2y = 5$ have a unique solu­tion.
10. A pair of equa­tions is $ax + by = c$ and $2ax + 2by = 2c$ with $a, b$ not both zero. With­out any cal­cu­la­tion, state how many solu­tions it has and why.

Answers#