Exercises 1 (Simultaneous Equations)

Solve the following equations by elimination and check your results:

  1. \[\begin{align*} 3x+4y &= 11\\ x+7y &= 15 \end{align*}\]

  2. \[\begin{align*} 2x+3y &= 16\\ 3x+2y &= 14 \end{align*}\]

Solve the following equations by substitution and check your results:

  1. \[\begin{align*} 5x+3y &= 29\\ \frac{3x}{4}-\frac{2y}{5} &= \frac{3}{10} \end{align*}\]

  2. \[\begin{align*} \frac{2x}{3}-\frac{y}{4} &= \frac{7}{12}\\ 4x+7y &= 37 \end{align*}\]

  3. \[\begin{align*} \frac{x}{4}+\frac{y}{5} &= \frac{3}{2}\\ \frac{2x}{7}-\frac{y}{4} &= \frac{5}{14} \end{align*}\]

  4. \[\begin{align*} \frac{x}{2}+\frac{y}{3} &= \frac{13}{6}\\ 2x+3y &= 19 \end{align*}\]

  5. During a lab experiment on force resolution, the following equations were found: \[\begin{align*} 9F_H - 1.5F_V &= 7.5\\ 6.25 F_H - 2.5 F_V &= 8.75 \end{align*}\] Determine the values of both the horizontal and the vertical force components and check your results.

  6. A weight being moved against a frictional force is related by the law \[F=aW+b,\] where \(a\) and \(b\) are constants. When \(F=6\), \(W=7.5\), and when \(F=2.7\), \(W=2\). Determine the value of the constants \(a\) and \(b\) and check your results.

Solve by substitution:

  1. \[\begin{align*} y &= 2x\\ 3x+2y &= 21 \end{align*}\]

  2. \[\begin{align*} y &= 3x-7\\ 5x-3y &= 1 \end{align*}\]

  3. \[\begin{align*} x &= 5y-3\\ 3x-8y &= 12 \end{align*}\]

  4. \[\begin{align*} 2x -y &= 10\\ 3x+2y &= 29 \end{align*}\]

  5. \[\begin{align*} \frac{y}{2}-x &= 2\\ 6x-\frac{3y}{2} &= 3 \end{align*}\]

  6. \[\begin{align*} \frac{x}{2}-\frac{y}{3} &= \frac{1}{6}\\ \frac{y}{2}-\frac{x}{6} &= 5 \end{align*}\]

  7. The cost of 4 ties and 6 pairs of socks was £68, while that of 5 ties and 8 pairs of socks was £87.40. What were the prices of a tie and a pair of socks, respectively?

  8. The bill for the telephone for a quarter can be expressed in the form \[C = a+ \frac{nb}{100},\] where \(C\) is the total cost in pounds, \(a\) is the fixed charge, \(n\) is the number of calls, and \(b\) is the price of each call in pence. When 104 calls were made, the bill was £58.30, and when 67 calls were made, the bill was £50.90. Find the fixed charge and the cost of each call.

Solutions: 1. \(x=1,y=2\); 2. \(x=2,y=4\); 3. \(x=2.941, y=4.765\); 4. \(x=2.353,y=3.941\); 5. \(x=3.731, y=2.836\); 6. \(x=0.2,y=6.2\); 7. \(F_H=0.43, F_V=-2.43\); 8. \(a=0.6,b=1.5\); 9. \(x=3, y=6\); 10. \(x=5,y=8\); 11. \(x=12, y=3\); 12. \(x=7,y=4\); 13. \(x=3,y=10\); 14. \(x=9,y=13\); 15. tie £9.80, socks £4.80; 16. fixed charge £37.50, call 20p;