Pair of Linear Equations in Two Variables

The One Thing

A pair of linear equations in two variables represents two straight lines, and its solution is the common point of those lines. The pair has a unique solution, no solution, or infinitely many solutions according to whether the lines intersect, are distinct and parallel, or coincide.

Definitions and Results

• Linear Equation in Two Variables: An equation of the form $ax+by+c=0$, where $a$, $b$, and $c$ are real numbers and $a$ and $b$ are not both zero.

• Pair of Linear Equations: Two linear equations involving the same two variables, written in standard form as

$$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0.$$

• Solution: An ordered pair $(x,y)$ that satisfies both equations simultaneously.

• Graphical Method: Each equation is represented by a straight line on a coordinate plane. The point of intersection gives the common solution.

• Substitution Method: One variable is expressed in terms of the other and substituted into the second equation.

• Elimination Method: The equations are multiplied if necessary and then added or subtracted to eliminate one variable.

• Cross-Multiplication Method: For

$$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0,$$ the variables may be found using $$\frac{x}{b_1{c}_2-b_2{c}_1}=\frac{y}{c_1{a}_2-c_2{a}_1}=\frac{1}{a_1{b}_2-a_2{b}_1}.$$ If a denominator is zero, the ratio conditions must be examined before deciding the type of solution.

• Consistent Pair: A pair of equations with at least one solution. A consistent pair may have either a unique solution or infinitely many solutions.

• Inconsistent Pair: A pair of equations with no solution.

• Dependent Pair: A pair of equations representing the same line and therefore having infinitely many solutions.

• Unique Solution: Exactly one solution, occurring when the two lines intersect at one point. The algebraic condition is

$$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}.$$

• No Solution: No common solution, occurring when the two lines are distinct and parallel. The algebraic condition is

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}.$$

• Infinitely Many Solutions: Every point on one line lies on the other because both equations represent the same line. The algebraic condition is

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}.$$

• Graphical Interpretation: Intersecting lines give one solution, parallel distinct lines give no solution, and coincident lines give infinitely many solutions.

• Exact and Approximate Solutions: A graph may provide an approximate solution when the intersection does not lie exactly on a grid point, whereas algebraic methods generally provide an exact solution.

• Word Problems: A situation is converted into a pair of equations by assigning variables, translating the stated conditions into equations, solving the pair, and checking the result in the original context.

Worked Methods

Graphical Method

• Represent each equation as a straight line on a coordinate plane.
• Identify the relationship between the two lines.
• Read the common point, if one exists, as the solution.

The possible outcomes are:

• Intersecting lines: one solution.
• Parallel distinct lines: no solution.
• Coincident lines: infinitely many solutions.

If the intersection does not fall exactly on a grid point, the graph gives an approximate solution. An algebraic method can then be used to obtain the exact solution.

Substitution Method

• Solve one equation for one variable.
• Substitute this expression into the second equation.
• Solve the resulting equation for the remaining variable.
• Substitute that value back into the rearranged equation.
• Obtain the ordered pair $(x,y)$.
• Check the pair in both original equations.

This method is especially useful when one variable can be isolated readily.

Elimination Method

• Write the two equations in standard form.
• Choose a variable to eliminate.
• Multiply one or both equations if necessary so that the coefficients of the chosen variable are equal or opposites.
• Add or subtract the equations to eliminate that variable.
• Solve for the remaining variable.
• Substitute the value into either original equation to find the other variable.
• Check the resulting ordered pair in both original equations.

Elimination is often efficient when the coefficients are already suitable or can easily be made suitable.

Cross-Multiplication Method

• Express the pair in the standard form

$$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0.$$

• Apply

$$\frac{x}{b_1{c}_2-b_2{c}_1}=\frac{y}{c_1{a}_2-c_2{a}_1}=\frac{1}{a_1{b}_2-a_2{b}_1}.$$

• Determine $x$ and $y$ from the corresponding ratios.
• If any denominator is zero, do not conclude immediately. Check the coefficient ratios:

$$\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}.$$

• Verify the solution in both original equations.

Choosing and Checking a Method

• Use the graphical method when the geometric relationship between the lines is required.
• Use substitution when a variable is easily isolated.
• Use elimination when coefficients can readily be made equal or opposite.
• Use cross-multiplication when the equations are already in the required standard form.
• Check every algebraic solution by substitution into both original equations.

All valid methods must produce the same solution because they transform the original equations without changing their common solutions.

Word-Problem Method

• Assign variables to the unknown quantities.
• Translate the given conditions into two linear equations.
• Solve the resulting pair using an appropriate method.
• Interpret the ordered pair in the original situation.
• Check the result against the original conditions.

Where It Goes Wrong

• Failing to write both equations in the standard form

$$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0$$ before applying ratio tests or the cross-multiplication formula.

• Using the unique-solution condition without checking that

$$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}.$$

• Confusing distinct parallel lines with coincident lines: no solution requires

$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2},$$ whereas infinitely many solutions require equality of all three ratios.

• Treating a zero denominator in the cross-multiplication formula as an immediate conclusion about the solution type; the ratio conditions must first be checked.

• Omitting the back-substitution step in substitution or elimination, so that only one variable is found.

• Failing to substitute the final ordered pair into both original equations, particularly after solving a word problem.

What Gets Asked

This material supports questions requiring students to:

• Define a linear equation, a pair of linear equations, a solution, consistent and inconsistent pairs, and dependent pairs.
• Classify a pair as having a unique solution, no solution, or infinitely many solutions using coefficient ratios.
• Interpret pairs graphically as intersecting, parallel, or coincident lines.
• Solve a pair using the graphical, substitution, elimination, or cross-multiplication method.
• Identify the most efficient algebraic method from the form of the equations.
• Explain why algebraic methods give the same common solution.
• Handle zero denominators in the cross-multiplication formula correctly.
• Obtain an approximate graphical solution or an exact algebraic solution.
• Form and solve a pair of equations from a word problem.
• Check a proposed solution in both original equations and, where relevant, in the original situation.