### Subscribe Us # Straight Line Equations (CIE)

$\def\D{\displaystyle}\newcommand{\vcol}{\begin{array}{|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}\newcommand{\vicol}{\begin{array}{|c|c|c|c|c|c|}\hline#1\\ \hline#2\\ \hline\end{array}}$1 (CIE 2012, s, paper 11, question 7)
The table shows values of variables $\D x$ and $\D y.$
$\D \vicol{x& 1& 3& 6& 10& 14}{y& 2.5& 4.5& 0 &–20 &–56}$

(i) By plotting a suitable straight line graph, show that $\D y$ and $\D x$ are related by the equation $\D y = Ax + Bx^2,$ where $\D A$ and $\D B$ are constants. 
(ii) Use your graph to find the value of $\D A$ and of $\D B.$ 

2 (CIE 2012, s, paper 22, question 7)
The table shows experimental values of variables $\D x$ and $\D y.$
$\D \vcol{x& 5& 30& 150& 400}{y& 8.9& 21.9& 48.9& 80.6}$
(i) By plotting a suitable straight line graph, show that $\D y$ and $\D x$ are related by the equation $\D y = ax^b,$ where $\D a$ and $\D b$ are constants. 
(ii) Use your graph to estimate the value of $\D a$ and of $\D b.$ 

3 (CIE 2012, w, paper 11, question 10)
The table shows values of the variables $\D x$ and $\D y.$
$\D\vicol{ x^{\circ}& 10& 30 &45 &60 &80}{ y &11.2& 16 &19.5& 22.4& 24.7}$
(i) Using the graph paper below, plot a suitable straight line graph to show that, for 10° $\D \le x\le$ 80°, $\D \sqrt{y} = A \sin x + B,$ where $\D A$ and $\D B$ are positive constants. 
(ii) Use your graph to find the value of $\D A$ and of $\D B.$ 
(iii) Estimate the value of $\D y$ when $\D x = 50.$ 
(iv) Estimate the value of $\D x$ when $\D y = 12.$ 

4 (CIE 2012, w, paper 22, question 8)

The variables $\D x$ and $\D y$ are related in such a way that when $\D \lg y$ is plotted against $\D \lg x$ a straight line graph is obtained as shown in the diagram. The line passes through the points (2, 4) and (8, 7).
(i) Express $\D y$ in terms of $\D x,$ giving your answer in the form $\D y = ax^b,$ where $\D a$ and $\D b$ are constants. 
Another method of drawing a straight line graph for the relationship $\D y = ax^b,$ found in part (i), involves plotting $\D \lg x$ on the horizontal axis and $\D \lg(y^2)$ on the vertical axis. For this straight line graph what is
(iii) the intercept on the vertical axis? 

5 (CIE 2012, w, paper 23, question 9)
The table shows experimental values of two variables $\D x$ and $\D y.$
$\D \vcol{ x& 1& 2& 3& 4}{y& 9.41 &1.29& – 0.69& – 1.77}$
It is known that $\D x$ and $\D y$ are related by the equation $\D y = \frac{a}{x^2}+bx,$  where $\D a$ and $\D b$ are constants.
(i) A straight line graph is to be drawn to represent this information. Given that $\D x^2y$ is plotted on the vertical axis, state the variable to be plotted on the horizontal axis. 
(ii) On the grid opposite, draw this straight line graph. 
(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ 
(iv) Estimate the value of $\D y$ when $\D x$ is 3.7. 

6 (CIE 2013, s, paper 11, question 2)
Variables $\D x$ and $\D y$ are such that $\D y= Ab^x,$  where $\D A$ and $\D b$ are constants. The diagram shows the graph of $\D \ln y$ against $\D x,$ passing through the points (2, 4) and (8, 10). Find the value of $\D A$ and of $\D b.$ 

7 (CIE 2013, s, paper 22, question 1)

Variables $\D x$ and $\D y$ are such that when $\D \sqrt{y}$ is plotted against $\D x^2$ a straight line graph passing through the points (1, 3) and (4, 18) is obtained. Express $\D y$ in terms of $\D x.$ 

8 (CIE 2013, w, paper 13, question 10)
The variables $\D s$ and $\D t$ are related by the equation $\D t= ks^n,$ where $\D k$ and $\D n$ are constants. The table below shows values of variables $\D s$ and $\D t.$
$\D \vcol{s& 2& 4& 6& 8}{t& 25.00& 6.25& 2.78& 1.56}$
(i) A straight line graph is to be drawn for this information with $\D \lg t$ plotted on the vertical axis. State the variable which must be plotted on the horizontal axis. 
(ii) Draw this straight line graph on the grid below. 
(iii) Use your graph to find the value of $\D k$ and of $\D n.$ 
(iv) Estimate the value of $\D s$ when $\D t = 4.$ 

9 (CIE 2013, w, paper 21, question 8)
The table shows experimental values of two variables $\D x$ and $\D y.$
$\D \vcol{x& 2 &4& 6& 8}{y& 9.6& 38.4& 105& 232}$
It is known that $\D x$ and $\D y$ are related by the equation $\D y= ax^3+ bx,$ where $\D a$ and $\D b$ are constants.
(i) A straight line graph is to be drawn for this information with $\D \frac{y}{x}$ on the vertical axis. State the variable which must be plotted on the horizontal axis. 
(ii) Draw this straight line graph on the grid below. 
(iii) Use your graph to estimate the value of $\D a$ and of $\D b.$ 
(iv) Estimate the value of $\D x$ for which $\D 2y = 25x.$ 

10 (CIE 2014, s, paper 11, question 8)
The table shows values of variables $\D V$ and $\D p.$
$\D \vcol{ V &10& 50& 100& 200}{p& 95.0& 8.5& 3.0& 1.1}$
(i) By plotting a suitable straight line graph, show that $\D V$ and $\D p$ are related by the equation $\D p = kV^n ,$
where $\D k$ and $\D n$ are constants. 
(ii) the value of $\D n,$ 
(iii) the value of $\D p$ when $\D V = 35.$ 

11 (CIE 2014, s, paper 13, question 10)
The table shows experimental values of $\D x$ and $\D y.$
$\D \vcol{x& 1.50 &1.75& 2.00& 2.25}{y& 3.9& 8.3 &19.5& 51.7}$
(i) Complete the following table.
$\D \vcol{x^2&\qquad &\qquad &\qquad &\qquad}{\lg y&&&&}$

(ii) By plotting a suitable straight line graph on the graph paper, show that $\D x$ and $\D y$ are related by the equation $\D y= Ab^{x^2},$  where $\D A$ and $\D b$ are constants. 
(iii) Use your graph to find the value of $\D A$ and of $\D b.$ 
(iv) Estimate the value of $\D y$ when $\D x = 1.25.$ 

12 (CIE 2014, s, paper 22, question 10)
Two variables $\D x$ and $\D y$ are connected by the relationship $\D y = Ab^x ,$ where $\D A$ and $\D b$ are constants.
(i) Transform the relationship $\D y = Ab^x$ into a straight line form. 
An experiment was carried out measuring values of $\D y$ for certain values of $\D x.$ The values of $\D \ln y$ and $\D x$ were plotted and a line of best fit was drawn. The graph is shown on the grid below.

(ii) Use the graph to determine the value of $\D A$ and the value of $\D b,$ giving each to 1 significant figure. 
(iii) Find $\D x$ when $\D y = 220.$ 

13 (CIE 2014, w, paper 11, question 9)
The table shows experimental values of variables $\D x$ and $\D y.$
$\D \vicol{x& 2& 2.5& 3 &3.5& 4}{y& 18.8& 29.6& 46.9& 74.1 &117.2}$
(i) By plotting a suitable straight line graph on the grid below, show that $\D x$ and $\D y$ are related by the equation $\D y = ab^x ,$ where $\D a$ and $\D b$ are constants. 
(ii) Use your graph to find the value of $\D a$ and of $\D b.$ 

14 (CIE 2014, w, paper 23, question 6)
Variables $\D x$ and $\D y$ are such that, when $\D \ln y$ is plotted against $\D 3^x ,$ a straight line graph passing through (4, 19) and (9, 39) is obtained.

(i) Find the equation of this line in the form $\D \ln y= m3^x+ c,$  where $\D m$ and $\D c$ are constants to be found. 
(ii) Find $\D y$ when $\D x = 0.5.$ 
(iii) Find $\D x$ when $\D y = 2000.$ 

1. (i) $\D y/x = A + Bx$
$\D \vicol{x& 1& 3& 6& 10& 14}{y/x& 2.5& 1.5& 0& -2& -4}$
(ii) $\D B = -0.5;A = 3$
2. (i) $\D \ln y = ln a + b ln x$
(ii) $\D b = 0.5; a = 4$
(iii) 32 to 49
3. (i) $\D \vicol{\sin x& 0.17& 0.5& 0.71& 0.87& 0.98}{\sqrt{y}& 3.35& 4 &4.42& 4.73& 4.97}$
(ii) $\D A = 2;B = 3$
(iii) $\D y = 20.5$
(iv) $\D x = 14.5$
4. (i) $\D y = 1000\sqrt{x}$
(ii) $\D m = 1$
(iii) $\D c = 6$
5. (i) $\D x^3$
(ii) $\D \vcol{x^3& 1& 8& 27& 64}{x^2y& 9.41 &5.16& -6.21& -28.32}$
(iii) $\D a = 10; b = -0.6$
(iv) $\D -1.48$
6. $\D b = e;A = e^2$
7. $\D y = (5x^2 - 2)^2$
8. (i) $\D \lg s$
(ii) $\D \vcol{\lg s& 0.3 &0.6& 0.78& 0.9}{lg t& 1.4& 0.8& 0.44& 0.19}$
(iii) $\D n = -2; k = 100$
(iv) $\D s = 4.9$
9. (i) $\D x^2$
(ii) $\D \vcol{x^2& 4& 16& 36& 64}{\frac{y}{x}& 4.8& 9.6& 17.5& 29}$
(iv) $\D 4.8$
10. (i)
(ii) $\D n = 1.5$
(iii) $\D 15$
11. (i) $\D \vcol{x^2& 2.25& 3.06& 4& 5.06}{\lg y& 0.59& 0.92 &1.29& 1.71}$
(ii)
(iii) $\D b = 2.5;A = 0.5$
(iv) $\D 2.1$
12. (i) $\D \log y = \log A + x \log b$
(ii) $\D 0.5$ (iii) $\D 4.4$
13. (i)
(ii) $\D b = 2.5; a = 3$
14. (i) $\D \ln y = 4(3^x) + 3$
(ii) $\D y = 20500$
(iii) $\D x = 0.127$