Solve two equations at once to find where lines meet
Solve systems by graphing, substitution, elimination, modeling, and method selection.
Common CoreA-REI.C.5A-REI.C.6A-REI.D.12
7 lessons25 short videos6 key concepts, out loud
7.1
Solving Systems by Graphing
Find the solution as the intersection of two lines
A-REI.C.6
Key idea
A solution to a system of linear equations is the coordinate point (x, y) where the graphs of the equations intersect, representing the values that make both equations true at the same time.
Learn how to graph two linear equations on the same coordinate plane to find their single point of intersection.
Key concept · Lesson 7.1
The crossing point solves both
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Transcript. Graph both lines on the same axes. Wherever they cross is the one pair of values that satisfies both equations, so the answer is a coordinate pair, not a single number. Parallel lines never cross, meaning no solution, and identical lines share every point.
7.2
Solving Systems by Substitution
Swap one equation into the other
A-REI.C.6
Key idea
To solve a system of equations by substitution, replace a variable in one equation with its equivalent expression from the other equation to create a single-variable equation that can be solved.
Learn how to solve systems of equations algebraically by swapping a variable for an equivalent expression.
Key concept · Lesson 7.2
Put one equation inside the other
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Transcript. Substitution works when one variable is already alone. Take that expression and drop it into the other equation, in place of the variable, so only one letter is left. Solve, then put the value back to find the partner. Distribute carefully around what you swapped in.
7.3
Solving Systems by Elimination
Add or subtract equations to cancel a variable
A-REI.C.5A-REI.C.6
Key idea
To solve a system of equations using elimination, add or subtract the equations to cancel out one variable with opposite or matching coefficients, then solve for the remaining variable.
Learn how to align and combine equations to eliminate one variable, allowing you to solve for both unknowns quickly without graphing or substitution.
Key concept · Lesson 7.3
Cancel a column, then solve
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Transcript. Elimination lines the equations up in columns. If one column already has opposite coefficients, adding the equations makes it vanish. If the coefficients match instead, subtract, and distribute that minus sign to every term. When neither fits, multiply an equation first to force a match.
7.4
Systems Word Problems (Mixture / Motion)
Model and solve real situations
A-CED.A.3
Key idea
To model mixture and motion scenarios, translate the situation into a system of two equations—typically one representing total quantities and the other representing values, costs, or rates—and solve using substitution or elimination.
Learn how to set up and solve systems of equations for real-world scenarios like mixing solutions or tracking traveling vehicles.
Key concept · Lesson 7.4
Turn each total into an equation
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Transcript. Mixture and motion problems almost always need two equations of different kinds. One counts things: how many of each. The other measures their value, cost, or distance. Label both variables in words before you write anything, and never mix a count and a value inside one equation.
7.5
Systems of Inequalities
Shade the overlapping solution region
A-REI.D.12
Key idea
To solve a system of linear inequalities, graph both boundary lines—using dashed lines for strict inequalities and solid lines for inclusive ones—and shade the overlapping region where their individual solutions meet.
Learn how to graph two linear inequalities on the same coordinate plane and identify their shared solution region.
Key concept · Lesson 7.5
Shade both. Keep the overlap.
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Transcript. Graph each inequality as a boundary line: dashed when the sign is strict, solid when it allows equality. Then shade the side that works, testing a simple point if you are unsure. The solution is only the region where both shadings overlap, not either one alone.
7.6
Choosing the Best Method
Pick the fastest approach for a given system
Key idea
Choose the most efficient system-solving method by analyzing the equations' structure: use graphing for visual estimation, substitution when a variable is already isolated, and elimination when equations are in standard form.
Learn how to analyze the structure of a system of equations to pick the fastest, most error-free solving method.
Key concept · Lesson 7.6
Match the method to the shape
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Transcript. All three methods give the same answer, so pick the one that keeps the arithmetic clean. If a variable is already alone, substitute. If both equations sit in standard form, eliminate. Graphing is best when an estimate will do, since exact fractions are hard to read off a picture.
7.7
Review & Assessment
Systems, mixed practice
Review, practice, quiz and test
Key idea
Review systems with targeted mixed practice, a practice test, and a chapter test.
This is one chapter of the climb.
Algebra 1 spans 16 chapters, every lesson taught on video with a key concept you can hear, and Vector on hand to answer questions out loud.
