Number System & Expressions/Equations

Linear Equations and Systems of Equations

Grade 8

✓By the end of this lesson students will be able to solve linear equations with one variable, including those requiring the distributive property or with variables on both sides.
✓By the end of this lesson students will be able to understand what a system of two linear equations represents.
✓By the end of this lesson students will be able to solve systems of two linear equations in two variables using graphing, substitution, and elimination methods.
✓By the end of this lesson students will be able to interpret solutions to systems of linear equations in context.

Key concepts

Linear Equation (one variable)

An equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a ≠ 0. The goal is to find the value of the variable that makes the equation true.

ax + b = c

Solution to a Linear Equation

The value of the variable that makes the equation true. When you substitute the solution back into the original equation, both sides of the equation will be equal.

System of Linear Equations

A set of two or more linear equations with the same variables. For two equations with two variables (like 'x' and 'y'), we are looking for a pair of values (x, y) that satisfies both equations simultaneously.

ax + by = c\ndx + ey = f

Solution to a System of Linear Equations

An ordered pair (x, y) that satisfies all equations in the system. Graphically, the solution is the point of intersection of the lines represented by the equations.

Methods for Solving Systems

There are three primary methods for solving systems of linear equations: Graphing (plotting both lines and finding their intersection point), Substitution (solving one equation for one variable, then substituting that expression into the other equation), and Elimination (multiplying one or both equations by a constant to make the coefficients of one variable opposites, then adding the equations to eliminate that variable).

Key facts to remember

1To solve a linear equation, isolate the variable using inverse operations.
2The distributive property must be applied before combining like terms in an equation.
3A system of linear equations can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (the same line).
4The solution to a system of two linear equations is an ordered pair (x, y) that satisfies both equations simultaneously.
5Graphing, substitution, and elimination are the primary methods for solving systems of linear equations.
6Always check your solution by substituting the values back into the original equation(s) to ensure accuracy.

Worked examples

Example 1

Solve for x: 3(x - 2) + 5x = 18

IApply the distributive property: 3x - 6 + 5x = 18

IICombine like terms: 8x - 6 = 18

IIIAdd 6 to both sides of the equation: 8x = 24

IVDivide both sides by 8: x = 3

Answer

x = 3

Remember to use the distributive property first when parentheses are present.

Example 2

Solve the system of equations by substitution:\ny = 2x + 1\n3x + y = 11

IThe first equation is already solved for y: y = 2x + 1.

IISubstitute the expression (2x + 1) for y into the second equation: 3x + (2x + 1) = 11

IIICombine like terms: 5x + 1 = 11

IVSubtract 1 from both sides: 5x = 10

VDivide by 5: x = 2

VISubstitute x = 2 back into the first equation to find y: y = 2(2) + 1

VIISimplify: y = 4 + 1

VIIIy = 5

Answer

(2, 5)

Always check your solution by substituting the x and y values into *both* original equations.

Example 3

Solve the system of equations by elimination:\n2x + 3y = 7\n4x - 3y = 5

INotice that the coefficients of y are opposites (+3y and -3y).

IIAdd the two equations together to eliminate y:\n (2x + 3y) + (4x - 3y) = 7 + 5\n 6x = 12

IIIDivide both sides by 6: x = 2

IVSubstitute x = 2 into the first original equation: 2(2) + 3y = 7

VSimplify: 4 + 3y = 7

VISubtract 4 from both sides: 3y = 3

VIIDivide by 3: y = 1

Answer

(2, 1)

If no coefficients are opposites, multiply one or both equations by a constant to create opposite coefficients for one variable before adding.

Common mistakes

✗Incorrectly applying the distributive property, especially when a negative sign is involved outside the parentheses.
✗Forgetting to perform an operation on *both* sides of the equation, leading to an unbalanced equation.
✗Making arithmetic errors when combining like terms or performing inverse operations, which can lead to an incorrect solution.
✗When solving systems of equations, only finding the value for one variable (e.g., just x) and forgetting to find the other variable (y).
✗Not checking the solution in *both* equations of a system to verify it satisfies all conditions.

Exam tips

★Show all your work step-by-step to earn partial credit, even if your final answer is incorrect.
★For systems of equations, choose the method (graphing, substitution, or elimination) that seems most efficient for the given problem.
★If time permits, always check your solution by plugging the values back into the original equation(s) to catch any errors.
★Read the problem carefully to understand what is being asked, especially in word problems involving systems of equations.

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