edubun 한국어
Home›Algebra›Linear equations

Equations and solutions

In 2x + 1 = 7, only one number makes the two sides equal: 3, because 2 × 3 + 1 = 7.

What makes something an equation

An equation is a statement that two expressions have the same value. The sign between them, =, is the claim. Write 2x + 1 = 7 and you are saying that whatever number x stands for, doubling it and adding 1 gives 7. Some numbers make that claim true and others make it false. A number that makes it true is called a solution of the equation, and solving means finding every such number. OpenStax compares solving an equation to working out a puzzle: the solution is the answer, and the puzzle is not finished until you check that the answer fits.

The letters in an equation are called variables. They hold the place of a number we do not know yet. A number multiplying a variable is its coefficient: in 2x + 1 = 7 the coefficient of x is 2. A number on its own, such as the 1 and the 7, is a constant. In this course every equation is linear, which means the variable appears only to the first power. There is no x², no x under a square root and no x in a denominator. A linear equation in one variable usually has exactly one solution, and the next units build a short list of moves that finds it every time.

Checking a solution by substitution

You do not need to know how to solve an equation to test a possible answer. Replace the variable with the number, work out each side separately, and compare. For 2x + 1 = 7 and x = 3, the left side becomes 2 × 3 + 1 = 7 and the right side is 7, so the statement is true and 3 is a solution. Try x = 4 instead: the left side becomes 9, which is not 7, so 4 is not a solution. Keep the two sides apart while you work. A long chain such as 2 × 3 + 1 = 7 = 7 hides mistakes, while writing each side on its own line shows exactly where they agree or differ.

The graph lab below draws each side of an equation as a line. The blue line is y = 2x + 1, the left side, and the orange line is y = 7, the right side. They cross where both sides have the same value, and the x of that crossing point is the solution. Move the sliders for a, b, c and d and the crossing point moves with them. This picture returns in the unit on variables on both sides, and again in the graphs course, where two lines meeting is the whole idea of a system of equations.

Loading

Drag to turn. Scroll or pinch to zoom.

The equation ax + b = cx + d with a = 2, b = 1, c = 0, d = 7. The red mark on the x-axis is the solution, and the panel checks it by working out both sides.
BackgroundThe equals sign is younger than most people expect. Robert Recorde, a Welsh mathematician, introduced it in 1557 in his book The Whetstone of Witte. He wrote that he used a pair of parallel lines of one length because no two things can be more equal. Before that, writers spelled the idea out with a word such as aequales or an abbreviation. The word algebra is older: it comes from the title of al-Khwarizmi's book on calculation by completion and balancing, written in Baghdad in the ninth century. His al-jabr, completion, meant removing a subtracted term from an equation.
A solution is a number, not an equation. Write the answer as x = 3, and check it in the original equation rather than in a line you rewrote, because a copying slip in a later line would agree with itself.
EquationA statement that two expressions are equal, such as 2x + 1 = 7
SolutionA value of the variable that makes the equation true
VariableA letter that holds the place of an unknown number
CoefficientThe number multiplying a variable: 2 in 2x
ConstantA number standing alone, with no variable
LinearThe variable appears only to the first power
  1. Is x = 3 a solution of 2x + 1 = 7?
    • Yes
    • No
  2. Is x = 5 a solution of 3x − 4 = 10?
    • Yes
    • No
  3. Which value solves x + 8 = 15?
    • 7
    • 8
    • 23
    • 15
  4. What is the coefficient of y in 5y − 2 = 18?
  5. Is y = −2 a solution of 4y + 9 = 1?
    • Yes
    • No
  6. Which of these is a linear equation?
    • 3x + 2 = 11
    • x² = 9
    • 1/x = 4
    • √x = 5
  7. Is a = 1/2 a solution of 6a − 1 = 2?
    • Yes
    • No
  8. Which number solves 7 − x = 10?
    • −3
    • 3
    • 17
    • −17
  9. The lines y = 3x − 2 and y = 10 cross at one point. What is its x-coordinate?
  10. Is x = 6 a solution of 2(x − 1) = x + 4?
    • Yes
    • No
  11. Who first wrote the equals sign, in 1557?
    • Robert Recorde
    • René Descartes
    • Isaac Newton
    • al-Khwarizmi
Key points
  • A solution makes both sides of the equation equal.
  • Check a number by substituting it and working out each side separately.
  • In a linear equation the variable appears only to the first power.
  • Graphing each side as a line shows the solution where the lines cross.