In scientific experiments and mathematical modelling, many relationships between two variables are non-linear, forming curves rather than straight lines. By applying linear law, most non-linear equations can be converted into the linear form \(Y = mX + c\). Converting non-linear relations into linear form allows a straight line of best fit to be drawn, making it significantly easier to find the values of unknown constants from the gradient and the vertical intercept.
A visual summary showing how polynomial, rational and exponential equations are algebraically manipulated and compared to \(Y = mX + c\).
\(y = ax + \dfrac{b}{x} \xrightarrow{\times x} xy = ax^2 + b\)
Comparing with \(Y = mX + c\):
\(Y = xy,\quad X = x^2,\quad m = a,\quad c = b\)
\(y = pq^x \xrightarrow{\log_{10}} \log_{10} y = (\log_{10} q)x + \log_{10} p\)
\(Y = \log_{10} y,\quad X = x,\quad m = \log_{10} q,\quad c = \log_{10} p\)
Standard Linear Form
\(X\) and \(Y\) contain variables only; \(m\) and \(c\) contain constants only.
Linear law provides an algebraic bridge to turn curves into straight lines.
Non-linear relations can be converted into linear relations of the form \(Y = mX + c\) so that a straight line can be drawn. It is easier to extract information, such as constants and rates, from a straight line graph than from a curve.
\(X\) and \(Y\) must contain variables only; they cannot contain unknown constants. \(m\) (the gradient) and \(c\) (the vertical intercept) must contain constants only; they cannot contain variables.
For \(y = ax + \dfrac{b}{x}\), multiply both sides by \(x\) to get \(xy = ax^2 + b\), or divide both sides by \(x\) to get \(\dfrac{y}{x} = b\left(\dfrac{1}{x^2}\right) + a\).
For \(y = pq^x\), take \(\log_{10}\) of both sides: \(\log_{10} y = (\log_{10} q)x + \log_{10} p\). For \(y = \dfrac{p^x}{q}\), \(\log_{10} y = (\log_{10} p)x - \log_{10} q\).
If the equation involves powers of variables in polynomial or rational form, multiply or divide throughout by \(x\) or powers of \(x\).
If the equation involves index or exponential terms, apply logarithms to both sides and use the laws of logarithms.
Group the transformed equation into the structure \(Y = mX + c\). Ensure \(Y\) and \(X\) contain only variables, while \(m\) and \(c\) contain only constants.
After plotting \(Y\) against \(X\), read the intercept \(c\) and calculate the gradient \(m\). Substitute these into your algebraic expressions to solve for the constants.
Gradient of Line of Best Fit
Use two reference points on the line of best fit.
Logarithmic Laws Used
\(\log_{10}\left(\dfrac{a}{b}\right) = \log_{10} a - \log_{10} b,\quad \log_{10}(a^b) = b \log_{10} a\)
Answer in your mind, then press "Check Answer".
Convert the non-linear equation \(y = px^2 - q\) into the linear form \(Y = mX + c\). State \(Y\), \(X\), \(m\), and \(c\).
Convert the equation \(y = ax^3 + bx^2\) into linear form such that the horizontal axis represents \(x\). State \(Y\), \(m\), and \(c\).
Given the non-linear equation \(y = \dfrac{p^x}{q}\), express it in linear form using base-10 logarithms. If the line of best fit of \(\log_{10} y\) against \(x\) has a vertical intercept of \(-2.5\), calculate the value of \(q\).
There is only one unique way to convert any non-linear equation into linear form.
An equation can often be converted in more than one way. For example, \(y = ax + \dfrac{b}{x}\) can be converted into \(xy = ax^2 + b\) or \(\dfrac{y}{x} = b\left(\dfrac{1}{x^2}\right) + a\).
Logarithms must be used for every non-linear conversion.
Logarithms are necessary when the variable appears as an index/exponent, such as \(y = pq^x\), or when variables are related by power laws, such as \(y = ax^b\). Polynomial and rational relations are converted using basic algebraic operations like multiplying or dividing by powers of \(x\).
Objective: Convert non-linear equations into linear form and identify \(Y\), \(X\), \(m\), and \(c\).
1 Given the equation \(y = 5x^2 + 3x\), which of the following represents the equation in the linear form \(Y = mX + c\) when both sides are divided by \(x\)?
2 The variables \(x\) and \(y\) are related by the equation \(y = pq^x\). When converted into the linear form \(Y = mX + c\) using \(\log_{10}\), what are the gradient \(m\) and the vertical intercept \(c\)?
3 In the linear form \(Y = mX + c\), the variables \(X\) and \(Y\) must contain only variables, while \(m\) and \(c\) must contain only ________.
4 For the non-linear equation \(3y - px^2 = qx\), dividing both sides by \(3x\) yields the linear form \(\dfrac{y}{x} = \dfrac{p}{3}(x) + \dfrac{q}{3}\). If the vertical intercept of the line of best fit is 22, the value of \(q\) is ________.
Match each non-linear equation and its conversion method to its corresponding linear form \(Y = mX + c\). On a phone, tap a term first, then tap the matching box.
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