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# Linear Algebra Ch. 5

### Systems of Equations

TermDefinition
coefficient (numerical) A number multiplying a variable or product of variables. For example, −7 is the coefficient of −7x.
constant term A number that is not multiplied by a variable. In the expression 2x + 3(5 − 2x) + 8, the number 8 is a constant term. The number 3 is not a constant term, because it is multiplied by a variable inside the parentheses.
counterexample An example showing that a statement has at least one exception; that is, a situation in which the statement is false. For example, the number 4 is a counterexample to the statement that all even numbers are greater than 7.
Equal Values Method A method for solving a system of equations. To use the Equal Values Method, take two expressions that are each equal to the same variable and set those expressions equal to each other. For example, in the system of equations below, −2x + 5 and x − 1 eac
equivalent Two expressions are equivalent if they have the same value. For example, 2 + 3 is equivalent to 1 + 4. Two equations are equivalent if they have all the same solutions. For example, y = 3x is equivalent to 2y = 6. Equivalent equations have the same
Fraction Buster Method “Fraction Busting” is a method of simplifying equations involving fractions that uses the Multiplicative Property of Equality to rearrange the equation so that no fractions remain. To use this method, multiply both sides of an equation by the common deno
growth One useful way to analyze a mathematical relationship is to examine how the output value grows as the input value increases. You can see this growth on a graph of a linear relationship by using a growth triangle.
linear equation An equation in two variables whose graph is a line. For example, y = 2.1x − 8 is a linear equation. The standard form for a linear equation is ax + by = c, where a, b, and c are constants and a and b are not both zero. Most linear equations can be writt
point of intersection A point of intersection is a point that the graphs of two equations have in common. For example, (3, 4) is a point of intersection of the two graphs shown below. Two graphs may have one point of intersection, several points of intersection, or no points
solution The number or numbers that when substituted into an equation or inequality make the equation or inequality true.
standard form ax + by = c, where a, b, and c are real numbers and a and b are not both zero. When you are given this form, it is often useful to write an equivalent equation in y = mx + b form to find the line’s slope and y-intercept.
system of equations A set of equations with the same variables. Solving this finding one or more solutions that make each of the equations in the system true. The solution gives a point of intersection of the graphs of the equations in the system.
term A single number, variable, or the product of numbers and variables. In an expression, terms are separated by addition or subtraction signs.
variable A symbol used to represent one or more numbers.
y-intercept The point(s) where a graph intersects the y-axis. A function has at most one of these; a relation may have several. This is important because it often represents the starting value of a quantity in a real-world situation.
y = mx + b When two quantities x and y have a linear relationship, that relationship can be represented with an equation in this form. The constant m is the slope, and b is the y-intercept of the graph.
Created by: EMarshall8