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Key Concepts

5.1 Algebraic Expressions

  • Algebra is useful because it allows us to understand many situations in real life by modeling them with expressions.
  • Algebraic expressions are the building blocks of algebra. From algebraic expressions we can create algebraic equations.
  • Algebraic expressions are often simplified and evaluated using the four arithmetic operations.

5.2 Linear Equations in One Variable with Applications

  • Solving linear equations means discovering what the value of the variable in a linear equation represents in the given conditions.
  • When solving a linear equation, most often you will have one solution; however, a linear equation may have no solutions or infinitely many solutions.

5.3 Linear Inequalities in One Variable with Applications

  • Inequalities can be used when the possible values (answers) in a certain situation are numerous, or when the exact value (answer) is not known, but it is known to be within a range of possible values.
  • Linear inequalities can be represented using a number line or using interval notation.

5.4 Ratios and Proportions

  • A ratio is a comparison of two numbers. The ratio of two numbers aa and bb can be written as: aa to bb OR aa:bb OR the fraction aa/bb.
  • All fractions are ratios, but not all ratios are fractions. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.
  • When two ratios are equal, we say they are in proportion or are proportional.
  • Setting up proportions allows us to solve many various situations where three of the four values of the proportion are known.

5.5 Graphing Linear Equations and Inequalities

  • Linear equations can be represented graphically on a rectangular coordinate system.
  • Solving linear equations in two variables means finding the point where two lines intersect. There are three possibilities: The lines intersect at exactly one point; the lines do not intersect (they are parallel); or the lines intersect everywhere (they are the same line).
  • Solving linear inequalities in two variables means finding a region of possible answers. Every point in this region will make both inequalities true statements.
  • Plotting points is a standard way to help graph linear equations and linear inequalities.

5.6 Quadratic Equations with Two Variables with Applications

  • A quadratic equation is an algebraic equation where the highest power (degree) of the equation is two.
  • To solve a quadratic equation is to find the value(s) that when substituted in for the variables, will make the equation equal to zero.
  • There can be two, one, or no solutions to any quadratic equation.
  • There are several methods to solve a quadratic equation. These methods include factoring quadratic equations, graphic quadratic equations, using the square root method, and using the quadratic formula.

5.7 Functions

  • A relation is any set of ordered pairs (x,y)(x,y). All of the xx-values of the set are the domain, and all of the yy-values of the set are the range.
  • A relation is a function if each xx-value in the domain is assigned to exactly one element in the range. A yy-value in the range can have more than one xx-value assigned to it; but each xx-value can only be assigned to one yy-value.
  • For the function y=f(x),fy=f(x),f is the name of the function, xx is the domain value variable, and y=f(x)y=f(x) is the range value variable.
  • The vertical line test is a test that can be done on the graph of a relation to determine if it is a function.

5.8 Graphing Functions

  • Every linear function can be graphically represented by a unique line that shows all the solutions of the equation.
  • The points where the graph of a line intersects the xx-axis and yy-axis are called the intercepts of the line.
  • Most lines will have one xx-intercept and one yy-intercept. Only if the line is straight vertical (no yy-intercept) or straight horizontal (no xx-intercept) will it not have both intercepts. Note that a line that is straight vertical is not a function, but a line that is straight horizontal is a function.
  • Since any two points determine a straight line, any linear function can be graphed if both intercepts are known.
  • The slope of a linear function is the ratio of the vertical change divided by the horizontal change. It is often referred to as riserunriserun.
  • A formula for finding the slope of linear functions is y2y1x2x1,y2y1x2x1, for any two points of the linear function (x1,y1)(x1,y1) and (x2,y2)(x2,y2).

5.9 Systems of Linear Equations in Two Variables

  • To solve a system of linear equations means finding the point or points where the two linear equations intersect.
  • Two lines can intersect at one point, no points if they are parallel, or every point if they are the same equation.
  • Systems of linear equations can be solved by graphing, by using substitution, or by using the elimination method.

5.10 Systems of Linear Inequalities in Two Variables

  • To solve a system of linear inequalities means to find the area(s) where the points in that area make all the linear inequalities true.
  • Systems of linear inequalities can be solved by graphing the linear equations associated with the inequalities, then 'testing' points to see whether the values of the point make the equation true or not.

5.11 Linear Programming

  • Linear programming is a mathematical technique to solve problems involving finding maximums or minimums where a linear function is limited by various constraints.
  • An objective function is a linear function in two or more variables that describes the quantity that needs to be maximized or minimized.
  • In linear programming, a constraint is a restriction that affects the maximum or minimum values of an objective function.
  • Through the creation of objective functions and restraints, a linear system can be developed and solved through linear programming.
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