SOLVING EQUATIONS
Solving a Linear Equation
To solve an equation, do whatever is necessary to both sides to isolate the variable. To solve the equation 5 x − 12 = −2x + 9, first
get all the x’s on one side by adding 2x to both sides: 7 x − 12 = 9. Then add 12 to both sides: 7 x = 21. Then divide both sides by 7:
x = 3.
Solving “In Terms Of”
To solve an equation for one variable in terms of another means to isolate the one variable on one side of the equation, leaving
an expression containing the other variable on the other side of the equation. To solve the equation 3 x − 10y = −5x + 6y for x in
terms of y, isolate x:
Translating from English into Algebra
To translate from English into algebra, look for the keywords and systematically turn phrases into algebraic expressions and
sentences into equations. Be careful about order, especially when subtraction is called for.
Example:Celine and Remi play tennis. Last year, Celine won 3 more than twice the number of matches that Remi won. If Celine
won 11 more matches than Remi, how many matches did Celine win?
Setup: You are given two sets of information. One way to solve this is to write a system of equations—one equation for each
set of information. Use variables that relate well to what they represent. For example, use r to represent Remi’s
winning matches, and use c to represent Celine’s winning matches. The phrase “Celine won 3 more than twice Remi,”
can be written as c = 2r + 3. The phrase “Celine won 11 more matches than Remi,” can be written as c = r + 11.
Solving a Quadratic Equation
To solve a quadratic equation, put it in the “ax^2 + bx + c = 0” form, factor the left side (if you can) and set each factor equal to 0
separately to get the two solutions. To solve x^2 + 12 = 7x, first rewrite it as x^2 − 7x + 12 = 0. Then factor the left side:
Solving a System of Equations
You can solve for two variables only if you have two distinct equations. Two forms of the same equation will not be adequate.
Combine the equations in such a way that one of the variables adds or subtracts out. To solve the two equations 4 x + 3y = 8 and