Make it equal

The idea

Equating a linear polynomial in one variable to a constant gives a linear equation.

Two numbers

The sum of two numbers is 64. One number is 10 more than the other. Call the smaller number x. Then the larger number is x + 10, and the sum is x + (x + 10) = 2x + 10.

We know the sum is 64. What can we say about 2x + 10?

• It is equal to 64
• It is less than 64
• It is equal to 10

Balance it

On the left pan is 2x + 10, the sum of the two numbers. On the right pan is 64. Change x until the scale balances.

Polynomial = constant

When we equate a linear polynomial in one variable to a constant, we get a linear equation.

2x + 10 is a linear polynomial. Equating it to 64 gives the linear equation 2x + 10 = 64. Then 2x = 54, so x = 27. The numbers are 27 and 37, and 27 + 37 = 64.

Notes

Equating a linear polynomial in one variable to a constant gives a linear equation, such as 2x + 10 = 64.

Check yourself

Which of these is a linear equation?

The sum of two numbers is 30. One is 4 more than the other. If the smaller is x, then x + (x + 4) = 30. What is the smaller number?

Answer: 13

2x + 4 = 30, so 2x = 26 and x = 13. The numbers are 13 and 17.

You equate the linear polynomial 5x − 2 to the constant 18. What equation do you get?

In the equation 2x + 10 = 64, what is the value of 2x + 10 when x = 27?

Answer: 64

2 × 27 + 10 = 54 + 10 = 64, which is the constant on the right side. So x = 27 solves the equation.

• 3x + 5 = 20 — correct. Yes. A linear polynomial, 3x + 5, is equated to the constant 20.
• 3x + 5. This is only a polynomial. It is not equated to anything yet.
• x² + 5 = 20. x² + 5 has degree 2, so it is not linear.
• 5x − 2 + 18. There is no equals sign, so it is not an equation.
• 5x − 2 = 18 — correct. Yes. Put the polynomial on one side and the constant on the other.
• 5x = 18. The −2 belongs to the polynomial. It must stay on the left.