Checking Candidate Solutions by Substitution

What It Means to Check a Candidate

A proposed solution to a system is a list of values for its variables. To check that proposal, substitute the values into the equations and test whether each equation is true. This is a direct test: it does not require solving the system. If even one equation fails, the proposed list is not a solution of the system.

The order of the values matters. If the variables have been ordered as \(x,y\), then the candidate \((2,3)\) means \(x=2\) and \(y=3\), not \(x=3\) and \(y=2\). For variables ordered as \(x,y,z\), the tuple \((a,b,c)\) means \(x=a\), \(y=b\), and \(z=c\). Throughout this tutorial, all variables and candidate values are real numbers.

“Every equation” is essential. A candidate that satisfies most, but not all, of the equations is not a solution of the system. A useful habit is to keep one line of arithmetic for each equation and compare the evaluated left-hand side with that equation’s right-hand side. Do not stop after checking the first equation: a candidate can pass one equation and fail another.

The Substitution Test

Proof. Suppose first that \((a_1,\dots,a_n)\) is a solution. By the definition of a solution, substituting \(x_i=a_i\) into every equation makes every equation true; in particular, each left-hand side equals its right-hand side. Conversely, suppose that substituting \(x_i=a_i\) makes the left-hand side equal the right-hand side in every equation. Then every equation in the system is true for those values. By the definition of a solution, \((a_1,\dots,a_n)\) is a solution. Both implications use the requirement that all the equations pass the test. \(\square\)

The test gives a quick way to disprove a proposed solution: one failed equation is enough. If an equation evaluates to \(6=9\), for example, it is false, so the tuple cannot satisfy the whole system. This conclusion is only about that candidate tuple. It does not, by itself, tell us whether some different tuple might satisfy the system.

Substitution and Coefficient Matrices

A system of linear equations can be written using a coefficient matrix. For a system of \(m\) equations in \(n\) ordered variables, let \(\mathbf{A}\in\mathbb{R}^{m\times n}\) contain the coefficients, and let \(\mathbf{b}\in\mathbb{R}^m\) contain the right-hand sides in the same equation order. Write a candidate tuple as the column vector \(\mathbf{x}=\begin{bmatrix}a_1&\cdots&a_n\end{bmatrix}^{\top}\). The matrix product \(\mathbf{A}\mathbf{x}\) lists the evaluated left-hand sides, one per row.

Proof. Let row \(i\) of \(\mathbf{A}\) be \((a_{i1},\dots,a_{in})\), and let the \(i\)th entry of \(\mathbf{b}\) be \(b_i\). By the definition of matrix multiplication, entry \(i\) of \(\mathbf{A}\mathbf{x}\) is \(a_{i1}x_1+\cdots+a_{in}x_n\). This is the left-hand side of equation \(i\) after the coefficients are applied to the ordered variables. Thus \(\mathbf{A}\mathbf{x}=\mathbf{b}\) holds exactly when each of its \(m\) entries equals the corresponding \(b_i\), which is exactly when every equation passes substitution. Therefore the candidate satisfies every equation if and only if \(\mathbf{A}\mathbf{x}=\mathbf{b}\). \(\square\)

This matrix form is a compact way to perform the same test, not a different test. Row \(i\) checks equation \(i\); the entries of \(\mathbf{b}\) must remain in the same order as the equations. If a product gives even one entry different from the corresponding entry of \(\mathbf{b}\), the candidate fails that row’s equation.

Common Mistakes to Avoid

Changing the variable order. Suppose the ordered variables are \(x,y\), and the candidate is \((2,3)\). A tempting but incorrect check of \(x+2y=8\) is to use \(x=3\), \(y=2\), giving \(3+2(2)=7\), and then declare that the candidate fails. The tuple does not assign \(x=3\), \(y=2\): it assigns \(x=2\), \(y=3\). The correct substitution is \(2+2(3)=8\), so this equation passes. Write the variable order next to the tuple if there is any chance of confusion.

Dropping a negative sign. For \(x-y=1\) and the candidate \((2,3)\) in the order \(x,y\), the correct left-hand side is \(2-3=-1\), not \(2+3=5\). Since \(-1\ne1\), the equation fails. Keep the subtraction sign attached to the variable it precedes.

Checking only one equation. An equation that passes confirms only that this one equation is true for the candidate. To verify a solution, check every equation. To reject a candidate, record at least one equation whose evaluated left-hand side differs from its right-hand side.

A reliable check can be written as a short list: name the equation, substitute the values, simplify the left-hand side, and compare with the right-hand side. Keeping the checks separate makes it easier to find an arithmetic slip and to state precisely which equation accepts or rejects the candidate.