Solve systems of two linear equations · lesson

Substitution

💡 If one equation says what y equals, put that into the other equation in place of y.

If one equation is y = something, you can swap that something in for y in the other equation.

Now there is only one letter. Solve for x. Then put x back in to get y.

🧠 Let's work it out

Solve y = 2x and x + y = 15.

What do you think comes next?

💡 When both equations start with y =, set the right sides equal and solve for x.

If both equations say y = …, then the two right sides must be equal where the lines meet.

Set them equal and solve for x. Then put x into either equation to find y.

🧠 Let's work it out

Solve y = 3x − 1 and y = x + 5.

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💡 If adding the two equations cancels one letter, add them to solve for the other.

Sometimes one letter has opposite coefficients, like +y and −y. Add the equations, left to left and right to right. That letter cancels.

If the coefficients are the same, like +y and +y, subtract the equations instead.

🧠 Let's work it out

Solve x + y = 13 and x − y = 5.

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💡 Write one equation for each fact in the story, then solve the system.

Look for two facts, like a total count and a total cost. Each fact becomes one equation.

"When do they cost the same?" means set the two costs equal.

🧠 Let's work it out

Plan A costs $30 plus $4 a month. Plan B costs $12 plus $7 a month. When are they equal?

Set the costs equal: 30 + 4m = 12 + 7m.
Subtract 4m and 12 from both sides: 18 = 3m.
Divide by 3: m = 6 months.
Check: 30 + 24 = 54 and 12 + 42 = 54.
Count-and-total stories work the same way: 10 bikes and trikes with 26 wheels give b + t = 10 and 2b + 3t = 26.
Double the first (2b + 2t = 20) and subtract it from the second: t = 6 trikes.

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💡 Parallel lines never meet (no solution); the same line meets itself everywhere (infinitely many).

Most systems have one solution, where the lines cross.

If the lines have the same slope but different y-intercepts, they are parallel. They never meet. There is no solution.

If the two equations are really the same line, every point on it works. There are infinitely many solutions.

🧠 Let's work it out

y = x + 3 and 3y = 3x + 9. How many solutions?

Divide the second by 3: y = x + 3.
It is the same line, so there are infinitely many solutions.
Compare: y = 3x + 2 and y = 3x − 5 have the same slope and different intercepts, so they never meet and have 0 solutions.

What do you think comes next?