Systems of Linear Equations in Two Variables

Introduction

Basic Competencies

3.5 Explain systems of linear equations in two variables and their solutions in contextual problems.

4.5 Solve problems related to systems of linear equations in two variables.

Learning Objectives

After studying this material, students are expected to be able to:

1. explain the meaning of a linear equation in two variables,
2. explain the general form of a system of linear equations in two variables,
3. solve SLETV problems using graph, substitution, elimination, and mixed methods,
4. solve word problems related to systems of linear equations in two variables.

Material

1. Linear Equations in Two Variables

Observe the following equations.

1. $2x + 2y = 1$
2. $\dfrac{m}{n} - \dfrac{n}{2} = 5$
3. $5p + 6q = -20$

Each equation has two variables, and each variable has degree one. Therefore, they are examples of linear equations in two variables.

Note

A linear equation in two variables can be written in the general form $ax + by = c$ with $a,b,c \in \mathbb{R}$, $a \neq 0$, and $b \neq 0$.

The solution of a linear equation in two variables can be found by inspection or by algebraic operations.

Example

Determine the non-negative integer solutions $(x,y)$ that satisfy:

If $x=0$, then:

If $x=3$, then:

So, the solution set is:

2. Systems of Linear Equations in Two Variables

A system of linear equations in two variables is a collection of two or more linear equations that use the same variables.

General form

$$> \left\{ > \begin{aligned} > a_1x+b_1y&=c_1 \\ > a_2x+b_2y&=c_2 > \end{aligned} > \right. >$$

Examples:

1. $\left\{\begin{aligned}2x+y&=-5\\3x+5y&=-4\end{aligned}\right.$
2. $\left\{\begin{aligned}y&=3x-2\\2x+y&=8\end{aligned}\right.$

Solving SLETV

Systems of linear equations in two variables can be solved by:

1. graph method,
2. substitution method,
3. elimination method,
4. mixed method.

3. Graph Method

In the graph method, the solution is the intersection point of the two lines on the Cartesian plane.

Example

Find the solution set of:

For the equation $x+y=5$:

For the equation $x-y=1$:

The graph is as follows.

The two lines intersect at $(3,2)$. Therefore, the solution set is:

4. Substitution Method

The substitution method is done by expressing one variable in terms of the other, then substituting it into the second equation.

Example

Solve:

From equation $(1)$:

Substitute $y=5-x$ into equation $(2)$:

Substitute $x=2$ into equation $(1)$:

So, the solution set is:

5. Elimination Method

The elimination method removes one variable by making its coefficients equal first if necessary.

Example

Solve:

Add the equations to eliminate $y$:

Hence:

Now eliminate $x$ by multiplying the second equation by 2:

So:

Therefore, the solution set is:

6. Mixed Method

The mixed method combines elimination and substitution.

Example

Solve:

Make the coefficients of $x$ equal:

Subtract the equations:

Thus:

Substitute into equation $(1)$:

So, the solution set is:

7. Word Problem

Word problems involving SLETV are solved by first creating a mathematical model and then solving the resulting system.

Example

The price of five tables and eight chairs is Rp1,150,000.00, while the price of three tables and five chairs is Rp700,000.00. Determine the price of each table and chair.

Let:

• table price = $x$
• chair price = $y$

Then:

Multiply equation $(1)$ by 3 and equation $(2)$ by 5:

Subtract:

Substitute $y=50{,}000$ into equation $(2)$:

Therefore:

• table price = Rp150,000.00
• chair price = Rp50,000.00

Evaluation

1. The sum of two numbers is 28 and their difference is 12. The numbers are … A. 8 and 20
   B. 10 and 18
   C. 12 and 16
   D. 14 and 14

2. Ali is 30 years old now. Six years ago, Ali’s age was three times Budi’s age. Budi’s current age is … A. 8 years
   B. 10 years
   C. 14 years
   D. 24 years

3. Observe the following graph.

The equation that matches the graph is … A. $2x+3y=3$
B. $2x+y=9$
C. $2x+y=3$
D. $3x+y=2$

4. The solution set of $3x-2y=7$ and $2x+y=14$ is $\{(a,b)\}$. The value of $a+b$ is … A. 9
   B. 7
   C. 5
   D. 4

5. On a farm there are 13 animals consisting of chickens and goats. The total number of legs is 38. The number of goats is … A. 5
   B. 6
   C. 7
   D. 8

That concludes this material on systems of linear equations in two variables.