Equations And Inequalities

 In this article, we are going to learn about Equalities and Inequalities:

• Relationship between the number of points of intersection and the nature of solutions of a pair of simultaneous equations
• Related conditions for a given line to
  • intersect a given curve
  • be a tangent to a given curve
  • not intersect a given curve

This article is specifically written to meet the requirements of Secondary 3 Additional Mathematics.

Quadratic Formula

The general solution of the equation \(ax^2 +bx + c =0\)  where \(a\neq 0\),  are solved using the quadratic formula, \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)  where \(x\) is called the roots of the equation. We will be using this to understand the nature of the roots.

Discriminant And Nature Of Roots

A discriminant, \(b^2-4ac\), is a part of the quadratic formula which can be used to tell us the number of roots a quadratic equation has and nature refers to the type of roots that exist.

For the quadratic formula \(\begin{align} x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{align}\)

Solving Linear Simultaneous Equations

Question 1:

Solve these simultaneous equations.

\(\begin{align} y &= 2x-1 \\ y &= -x+5 \end{align}\)

Solution:

You can solve this using the Substitution Method.

\(\begin{align} y &= 2x-1 & ---- \;(1) \\ y &= -x+5 & ----\;(2) \end{align}\) 

Since, both equations equate to \(y\) hence you can substitute the value of (1) in (2)    

\(\begin{align} 2x - 1 &= -x+5\\ \implies\; 2x + x &= 1 + 5\\ \implies\qquad 3x &= 6 \\ \implies\qquad\;\; x &=2 \;\;\;\text{ is the root} \end{align}\)

Solving Linear & Non–Linear Simultaneous Equation

Question 2:

Solve these simultaneous equations

\(\begin{align}​​ y &= x^2 +2x -5 \\ y &= -x+5 \end{align}\)

Solution: 

You can solve this using the Substitution Method.

\(\begin{align}​​ y &= x^2 +2x -5 & ----\;(1)\\ y &= -x+5 & ----\; (2) \end{align}\)

Since both equations equate to \(y\) hence you can substitute the value of (1) in (2)

\(\begin{align} -x+5 &= x^2+2x-5\\ \implies x^2 +2x -5 + x -5 &= 0\\ \implies\qquad\quad x^2 +3x -10 &= 0\\ \\ \end{align}\)

\(\begin{align} \therefore\quad x &= {-(3) \pm \sqrt{3^2-4(1)(-10)} \over 2(1)} \\ \implies x &= {-3 \pm \sqrt{49} \over 2} \\ \implies x &=2 \qquad \text{or} \qquad -5 \end{align}\)

Thus, the discriminant \(b^2 -4ac = 49\) which is \(> 0\) and hence there are two real roots, \(2\) and \(–5\).

Question 3:

Solve these simultaneous equations

\(\begin{align}​​ y &= x^2 -11x +30 \\ y &= -x +5 \end{align}\)

Solution:

You can solve this using the Substitution Method.

\(\begin{align}​​ y &= x^2 -11x +30 & ----\;(1)\\ y &= -x +5 & ----\;(2) \end{align}\)

Since both equations equate to \(y\) hence you can substitute the value of (1) in (2)

\(\begin{align} x^2-11x+30 &= -x+5\\ \implies\;\; x^2 -11x +30 +x-5 &= 0\\ \implies\qquad\qquad x^2 -10x +25 &= 0 \end{align}\)

\(\begin{align}​​ \therefore\quad x &= {-(-10) \pm \sqrt{(-10)^2-4(1)(25)} \over 2(1)} \\ \implies x &= {10 \pm \sqrt{0} \over 2}\\ \implies x &=5 \end{align}\)

Thus, the discriminant is \(b^2 -4ac = 0\) which is \( = 0\) and hence there is one real distinct root, 5.

Question 4:

Solve these simultaneous equations

\(\begin{align}​​ y &= x^2 -x +6 \\ y &= -x +5 \end{align}\)

Solution:

You can solve this using the Substitution Method.

\(\begin{align}​​ y &= x^2 -x +6 & ----\;(1)\\ y &= -x +5 & ----\;(2) \end{align}\)

Since both equations equate to \(y\) hence you can substitute the value of (1) in (2).

\(\begin{align} x^2-x+6 &= -x+5\\ \implies x^2 -x +6 +x-5 &= 0\\ \implies\qquad\quad x^2 +0x+1 &= 0 \end{align}\)

\(\begin{align}​​ \therefore\quad\qquad x &= {-(0) \pm \sqrt{(0)^2-4(1)(1)} \over 2(1)}\\ \implies\qquad x &= {-(0) + \sqrt{-4} \over 2} \\ \implies\qquad x &= {-(0) - \sqrt{-4} \over 2} \end{align}\)

As the discriminant \(b^2 -4ac = -4\)  which is \(< 0\) and hence there are no real distinct roots or no solution possible.

Related Conditions For Determining The Number Of Points Of Intersection Of A Line And A Curve

When two equations are given in the form

\(\begin{align}​​ y &=px^2 +qx +r \\ y &= ax +b \end{align}\)

You can solve them using \(\begin{align}​​ x = {-b \pm \sqrt{b^2-4ac} \over 2a} \end{align}\)

Example 1:

Find the value of \(m\) for which the line \(y = mx -3 \) is a tangent to the curve \(\begin{align}​​ y=x+\frac {1}{x} \end{align}\).

Solution:

When we look at the question, it mentions a tangent to the curve \(y=x+\frac {1}{x}\) which means the curve gets cut only at one point. It means that \(b^2 -4ac = 0\) and there is only one real root.

\(\begin{align}​​ y &= mx -3 & ----\;(1)\\ y &=x+\frac {1}{x} & ----\;(2) \end{align}\)

\(\begin{align} \implies\qquad\qquad\quad\; mx-3 &= x+ \frac{1}{x}\\ \implies\qquad\qquad\quad\; mx -3 &= \frac{x^2+1}{x}\\ \implies\qquad\qquad\; mx^2-3x &= x^2 +1\\ \implies\; x^2+1 -mx^2 +3x &=0\\ \implies (1-m)x^2 +3x +1 &=0 \end{align}\)

Now we have  \(a=(1-m),\; b=3,\; c=1\)

We already know \(b^2 -4ac = 0\)

Hence, 

\(\begin{align}​​ \implies 3^2 -4(1-m)(1)&=0\\ \implies\qquad\qquad\;\;\, 1-m &=\frac{9}{4} \\ \implies\qquad\qquad\qquad\;\, m &=1-\frac{9}{4} \\ \implies\qquad\qquad\qquad\;\, m &=-\frac{5}{4} \end{align}\)

Question 5:

Find the value of \(m\) for which the line \(y = x+m\) is a tangent to the curve \(y = 2x^2 -1\).

Solution: 

When we look at the question, it mentions a tangent to the curve \(y = 2x^2 -1\), which means the curve gets cut only at one point. It means that \(b^2 -4ac = 0\) and there is only one real root.

Equating both equations with each other we get,

\(\begin{align} x+m &=2x^2 -1\\ \implies\;\;\, 2x^2 -x-m-1 &= 0\\ \implies 2x^2 -x-(m+1 ) &= 0 \end{align}\)

Hence, we obtain \(a=2,\; b=-1 \;\text{and} \; c=-(m+1)\)

\(\begin{align} \therefore\qquad(-1)^2 -4(2)(-m-1)&=0\\ \implies\qquad\qquad 1-8(-m-1) &= 0\\ \implies\qquad\qquad\quad\;\; 1+8m +8 &= 0\\ \implies\qquad\qquad\qquad\quad\; 8m +9 &= 0\\ \implies\qquad\qquad\qquad\qquad\quad\;\; m &= -\frac{9}{8}\\ \implies\qquad\qquad\qquad\qquad\quad\;\ m &= -1\frac{1}{8} \end{align}\)

Conclusion

In this article, we have observed the relationship between the number of points of intersection and nature of solutions of a pair of simultaneous equations as per the Secondary 3 Additional Mathematics syllabus in Singapore.

We have also studied the related conditions for a given line to

• intersect a given curve
• be a tangent to a given curve
• not intersect a given curve

Multiple examples and questions are also given to aid in understanding these concepts better. Keep learning! Keep improving!'