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Algebra 1 Unit 4: Systems of linear equations

A system of equations asks for values that satisfy every equation at once. Graphically, it's where the lines cross.

Covers: Graphing, substitution, elimination, number of solutions, systems of inequalities, word problems

Study guide

A system of linear equations is two or more equations with the same variables. Its solution is the point that makes every equation true, which on a graph is where the lines intersect. Graphing shows the idea but is only exact when the intersection lands on grid points.

Substitution works best when a variable is already alone: solve one equation for a variable, substitute that expression into the other equation, solve, then plug back in to find the second variable. Elimination works best when coefficients line up: multiply one or both equations so a variable has opposite coefficients, add the equations to cancel it, and solve.

A system can have one solution (lines with different slopes cross once), no solution (parallel lines with the same slope and different intercepts; algebra gives a false statement like 0 = 5), or infinitely many solutions (the same line; algebra gives 0 = 0).

Systems of inequalities are solved by graphing each inequality, using a dashed boundary for and a solid one for ≤ or ≥, and shading the correct side. The solution is the region where all the shading overlaps. In word problems, define two variables, write one equation for each piece of information (often a total count and a total value), and solve.

To solve a linear-quadratic system algebraically, set the two expressions for y equal (or substitute the line into the quadratic), move everything to one side and solve the resulting quadratic. Find the y-value for each x, giving up to two intersection points; graphically, the line can cross the parabola twice, touch it once or miss it.

Key ideas

• Substitution works well when one variable is already isolated.
• Elimination works well when coefficients line up or can be scaled to match.
• Parallel lines → no solution. Same line → infinitely many.
• For inequalities, shade each region; the overlap is the solution set.
• Word problems: define variables, write one equation per fact.

Formulas and rules

Common mistakes

• Only finding one variable.
• Arithmetic errors when multiplying an equation to eliminate.

Words to know

Solution of a system (graph)

The point where the lines intersect.

Substitution method

Solve one equation for a variable and plug it into the other.

Parallel lines (system)

Same slope, different intercepts: no solution.

Solution region of a system of inequalities

Where all shaded regions overlap.

Independent equations

You need as many independent equations as unknowns.

Independent system (lines)

Different slopes: exactly one solution.

System of linear equations

Two or more linear equations with the same variables.

Elimination method

Add or subtract equations to cancel a variable.

Graphing method

Graph both lines; the intersection is the solution.

Inconsistent system

No solution; algebra gives a false statement like 0 = 5.

System of inequalities

Graph each inequality; the solution is the overlapping region.

Dashed vs. solid boundary

Dashed for ; solid for ≤ or ≥.

System word problems

Define two variables and write one equation per fact (often a total count and a total value).

Sample practice questions

25 more questions like these, from level 1 to 5, are in the practice set.