Solve systems algebraically and graphically · lesson

Substitution

💡 When one equation already isolates a variable, substitute that expression into the other equation.

If an equation reads y = … or x = …, replace that variable in the other equation with the expression.

Solve the one-variable equation. Then substitute back to find the other variable.

🧠 Let's work it out

Solve y = 2x + 1 and 3x + y = 16.

What do you think comes next?

💡 If one variable has opposite coefficients, add the equations to eliminate it.

Line the equations up, x under x and y under y.

If one variable has opposite coefficients, like +y and −y, add the equations. That variable drops out.

Answers can be fractions or decimals. Type them as they come, like (2.5, 1).

🧠 Let's work it out

Solve 2x + y = 11 and 3x − y = 4.

What do you think comes next?

💡 Multiply one equation so a variable's coefficients become opposites, then add.

If no variable cancels yet, multiply one equation (every term) by a number that makes the coefficients opposite.

Then add the equations as before.

🧠 Let's work it out

Solve x + 3y = 11 and 2x − y = 1.

What do you think comes next?

💡 Turn the count fact and the value fact into two equations, then solve.

Many stories give a total count and a total value. Use one variable for each kind.

Count: a + c = total items. Value: price × a + price × c = total cost. Solve the system.

🧠 Let's work it out

Adult tickets cost $9 and child tickets $4. A group buys 12 tickets for $78. How many adult tickets?

What do you think comes next?

💡 If elimination wipes out both variables, a false statement means no solution and a true one means infinitely many.

Sometimes both variables cancel. Look at what's left.

A false statement, like 0 = 5, means the lines are parallel. There is no solution.

A true statement, like 0 = 0, means the equations are the same line. There are infinitely many solutions.

🧠 Let's work it out

3x + 6y = 9 and x + 2y = 4. How many solutions?

What do you think comes next?