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7.6 Chapter summary

  • An equation is a mathematical sentence that is true for some numbers, but false for other numbers. For example, \(2x + 10 = 20\) will be true for \(x = 5\).
  • When we look for a number or numbers that make an equation true, we say that we are solving the equation or looking for a solution. A solution to an equation is a number that makes the equation true when you substitute it for the variable. For example, \(x = 10\) is the solution to the equation \(- 5x = - 50\).
  • When a certain number is the solution of an equation, we say that the number satisfies the equation.
  • Everything on the left-hand side of the equal sign must be equal to everything on the right-hand side of the equal sign, just like a balance scale.
  • Searching for the solution of an equation by using tables or by narrowing down to the possible solution is called solution by inspection. When solving by using inspection, we look at each side of the equation, and try to match the two sides up exactly.
  • When we solve equations, we are looking for the value of the unknown variable that will make the left-hand side equal to the right-hand side.
  • We name equations by looking at the highest power of \(x\) in the equation:
    • Equations with \(x\) terms only are called linear. For example, \(y = 2x + 1\).
    • Equations with \(x^{2}\) terms are called quadratic. For example, \(y = x^{2}\).
    • Equations with \(x^{3}\) terms are called cubic. For example, \(y = 2x^{3}\).