Solving Linear Equations: Part I

Solving Linear Equations YouTube Playlist

One method for solving a linear equation that looks like 2x + 3 = 13 is to simply guess and check.  What value can I substitute in for x so that the result will be 13?  After some thought it should be easy to see that x must equal 5 because 2 * 5 + 3 = 10 + 3 = 13.  However, guessing at solutions to linear equations becomes very difficult when the numbers are not so nice.  When solving, the idea is to use the properties of equality to isolate the variable.

Tip: We can always check by substituting our solution into the original equation to see if we obtain a true statement.  If you have time on an exam you should always check.

Solve for x.

Word Problem: Maria scored 7 more points than Jose on the first mathematics exam.  If Maria's total score was 95 what was Jose's total. (Set up an algebraic equation and solve it)

Tip: You can save a step if you multiply both sides of the equation by the reciprocal of the coefficient.

When solving, apply the addition or subtraction property of equality first. Next apply the multiplication or division property of equality as needed to isolate the variable.

Instructional Video: The Difference Between an Expression and an Equation

Solving General Two Step Equations
Steps for solving general linear equations of the form $ ax + b = c$.

$$ 2x + 3 = 13 $$

Add $-3$ on both sides of the equation.

$$ 2x = 10 $$

Divide both sides by $2$.

$$ x = 5 $$

Check: $2(5) + 3 = 10 + 3 = 13$ ✔

Solve for x.

Video Examples on YouTube:








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Graph by Plotting Points

Graphing lines can be done in a number of ways.  This section describes a method, plotting points, that always works and can be used for many other types of equations. Notice that in a linear equation with two variables, y = 3x - 2, the y-value depends on what the x-value is. Since x is independent here, choose any real number, say x = 4, and you can find the corresponding y-value by evaluating y = 3(4) – 2 = 12 – 2 = 10. Therefore, the ordered pair (4, 10) is a point on the graph of the equation.

[ Free Printable Graph Paper ]

Example: Graph y = −2x + 6 by plotting five points.

   Step 1: Choose any five x-values.
   Step 2: Evaluate to find the corresponding y-values.
   Step 3: Plot the points and use a straight edge to draw a line through them.

   

When choosing x-values it is wise to pick some negative numbers as well as zero.  Try to find the points where the line crosses the x and y axes.  These special points are called the x- and y-intercepts.

Is the given point a solution?

Remember that we are less likely to make a mistake if we insert parentheses where we see a variable and then substitute in the appropriate values.

Find the corresponding value.

Graph by plotting points.

Example: Graph y = 2x − 4 by plotting five points.

           

Example: Graph y = −x − 2 by plotting five points.

           

Choosing a scale when creating a blank coordinate system will take some thought.  Keep in mind that the scale on the x-axis need not be the same as the scale on the y-axis.

Example: Graph y = 1/2 x − 6 by plotting five points.

           

When the coefficient of x is a fraction, choose x-values to be multiples of the denominator so that you might avoid unnecessarily tedious calculations.

Example: Graph y = −3/2 x + 6 by plotting five points.

          

Example: Graph 2x − 3y = 6 by plotting five points.

          

When dividing a binomial by a number you must divide both terms by that number.  For example, treat the −3 as a common denominator as in the previous problem.

A common error is to just divide the 6 by −3.

Video Examples on YouTube: