Linear Equations – 1 Variable, 2 Variables, And Graphs

Linear Equations – 1 Variable, 2 Variables, And Graphs

A linear equation is a mathematical equation of degree 1. The degree of an equation is the highest degree of its terms while the degree of the terms in the equation is the sum of the exponents of the variables in it.

For example, the degree of the equation 2x + 3y = 5 is 1 because the highest power of any variable in the equation is 1. However, the degree of the equation 2xy - 3x - y = 1  is not linear (or 1) as the sum of the powers to which the variables x, y in the term 2xy are raised to 1 + 1 = 2.

Have a look at a few more examples of linear equations:

  • -2x + 3y = 1
  •  x - 1 = 2.5
  • -2x/3  - 3y/4 = 5

The linear equations questions in the math section of the SAT can be classified into 1 variable (linear) equations, 2 variables (quadratic) equations, wordy questions, and graphs of linear equations. For each of them, we shall understand the concepts and also look at a few examples.

Adding, subtracting, multiplying and dividing by terms on both sides of the linear equation are operations that one would have learnt in school days.  These operations are tested by SAT very often.

 

Linear equations in 1 variable:

A linear equation in 1 variable, as a concept, is self-explanatory. It is an equation in a single variable with its highest degree being 1.

Examples: 2x - 3 = 7

                -x + 2.5 = -3

Example:  Solve for x if 3.5 = 2.5x + 1.

Solution:

 Subtracting 1 on both sides of the equation, 3.5 - 1 = 2.5x + 1 - 1, which implies 2.5 = 2.5x. Now, dividing by 2.5 on both sides, we get 1 = x. We solved for x by isolating it from the equation.

The point is, if the questions are so simple, is there anything challenging in the SAT at all?

Yes, all the math questions in the SAT essentially test basic school-level mathematics. However, in a time bound test, there could be challenges hidden in simple questions as well.

For example, the question above may also ask to calculate the value of –x+1 instead of asking you to find the value of x. A lot of test takers end up choosing the wrong answer in such cases, mostly because it’s a time bound test where they often don’t take care of what is asked to be found/calculated and as they are so tuned to find the values of only x all through their school-level mathematics tests.

You may want to try this one to understand it better (though it’s not a linear equation problem):

If x / √2 = √50 / x , then what is the value of x - 1?              

(The above question was only checking whether you are very careful about the tricky nature of SAT questions)

Linear Equations in 2 variables:

A linear equation in 2 variables is an equation of degree 1 with the highest power of each variable being 1. Examples:

2x + 3y = 5

3x/2 - y = 0.5

Generally, these are equations that one would have seen simultaneously (2 at a time), where solving for the values of the 2 variables involved processes like elimination or substitution.

Example:  Solve the above 2 linear equations for x and y (in the example above).

Solution:  Elimination process involves multiplying the equation (2x + 3y = 5) with 3/4 so that the coefficients of x in both the equations are equal to each other.

3/4 (2x + 3y = 5)  Implies 3x/2 + 9y/4 = 15/4

Now, as the coefficients of x in both the equations are same, we can subtract one equation from another to eliminate x from them.

3x/2 + 9y/4 = 15/4 Implies

 9y/4 + y = 15/4 - 0.5

From which it can be implied that

13y/4=13/4.

Hence y = 1. Substituting the value of y back into any of the equation we also get the value of x as 1.

The challenges in questions on the SAT math on this concept are generally about taking care of the signs in the equations. Most of the questions that test takers go wrong at are those in which care has to be taken in handling the signs (-, +) when doing the operations explained above.

Wordy Questions:

The above two concepts of linear equations in 1 variable and 2 variables are tested in the formof questions cloaked in tricky sentences. The equations are generally simple translations of not-so-simple wordy situations in the questions. Translating the wordy questions to mathematical equations is generally tricky on the SAT.

Example: If twice a number is three less than itself, what is nine more than three times the number?

Solution:  Such wordy questions are better dealt in parts:

Firstly, assume the number to be               x

Then, twice the number would be             2x

Three less than itself would be                  x-3

          Given both are equal                                   2x = x-3

           Solving,                                                          x = -3

          Three times the number would be             3*(-3 ) = -9

          And nine more than it would be                 9+(-9) = 0

          Hence, the answer is                                     0

The math part on the right column of the above solution is all simple solving of a linear equation in 1 variable.

The wordy part on the left column of the above solution is all translating the wordy questions part by part into mathematical expressions/equations, which is what makes the simple math a bit more challenging in the SAT.

 

Example: If the price of 3 type A pencils less than the price of 2 type B pencils equals $1 and a set of 2 type A pencils and 3 type B  pencils costs  $8, what is the price of a type A  pencil?

Solution:  Solving a pair of linear equations in 2 variables: 2y - 3x = 1, 3y + 2x = 8 would give the answer.

How are the equations formed out of the wordy questions is what the SAT is testing one on.

So let’s understand the same:

Assume the price of one type A, type B pencil to be $x, $y respectively.

The price of 3 type A pencils would be 3 times x = 3x.

The price of 2 type B pencils would be 2 times y = 2y.

Also, the price of 3 type A pencils less than the price of 2 type B pencils would be = 2y - 3x.

Given, the price of 3 type A pencils less than the price of 2 type B pencils equals $1.

Hence, 2y - 3x = 1.

 

Similarly converting the information into equations we get another equation in 2 variables, 2x + 3y = 8.

To get the price of a type A pencil, we can simply solve for x using the pair of equations derived above.

x, y turns out to be 1 and 2 respectively.

 

You may try:

Albert’s and Benjamin’s ages add up to 35 years now. If Albert is twice as old as Benjamin was when Albert was as old as Benjamin is now, then how old is Albert?

Graphs:

All of the above linear equations can be graphed as lines on the coordinate plane (rectangular coordinate system).

Example: x/3 = -x + 1 is a linear equation in 1 variable that can be simplified to x = 3/4

On the coordinate plane, the graph of the above linear equation would look like as shown below:

 

Similarly, a linear equation in 2 variables, 2x + 3y = 6 can be plotted on the coordinate plane as shown below:

In general, the process to plot a graph manually would be to find the points that lie on the graph/equation and join them. In the second example, if we substitute x with 3, y would equal 0 and if we substitute x with 0, y would equal 2. Hence, the paired coordinates (3,0) and (0,2) lie on the line which can be seen in the graph too.

A table of paired values of x, y which satisfy the equation is shown below:

X

-3

-2

-1

0

1

2

3

y

4

10/3

8/3

2

4/3

2/3

0

 

All the points that lie on the line (graph) that can be derived from the above table would hence be (-3,4), (-2,10/3), (-1,8/3), (0,2), (1,4/3), (2,2/3) and (3,0) lie on the line shown in the graph.

A line containing points in which the value of y decreases as the value of x increases is negatively sloped (like the one graphed above) and shows a negative correlation between the 2 variables.

Plotting such a linear graph for another linear equation in 2 variables will help us understand the solution for the pair of linear equations if any.

 

If the lines graphed intersect, the point of intersection itself is the solution of the pair of linear equations. If the lines so plotted turn out to be parallel to each other, there exists no point of intersection and if the lines so plotted turn out to be coinciding(one over another), there exist infinitely many solutions.A pair of linear equations which do not have a solution (parallel lines) is shown below:

The equations 2x + 3y = 6  and 4x + 6y = 10  are plotted as shown and are hence parallel and do not have a solution (point of intersection).

 

 

The equations 2x + 3y = 6  and 4x + 6y = 12  are plotted as shown and are hence coinciding and have infinitely many solutions.

 

In general, the standard format of a linear equation is ax + by +c = 0 where x, y are variables and a, b, care constants.

If ax +by + c = 0 and px + qy + r = 0 is a pair of linear equations in 2 variables, they would have

A unique solution: If a/b is not equal to p/q.

No solution:  If a/b = p/q but not equal to c/r.

Infinitely many solutions:  If a/p = b/q = c/r.

 

You may try:

If 2x - 3y = 5 and px - 6y = r are two linear equations which have infinitely many solutions, what is the value of r-p?

Try to solve this question on your own! Do post your comments and solutions in the comment section below.

 

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