Systems of Linear Equations

Topic 1.1 Systems of Linear Equations	SLT
Video Please watch the video on Systems of Linear Equations.	8 mins
Notes 1.1 Please download and read Notes 1.1	30 mins
Activity 1.1 Please complete Activity 1.1	60 mins

Activity 1.1

Let's explore some questions and solutions related to systems of linear equations.

Question 1:

System of Equations: Solve the following system of linear equations:

$2x+y=5$

  

$3x-2y=8$

 

Solution: To solve the system, we can use methods like substitution or elimination. Let's use the substitution method for this example.

From the first equation, solve for y:

$y=5-2x$

  

Substitute this expression for 

$y$

into the second equation:

$3x-2(5-2x)=8$

3x−2(5−2x)=8

Simplify and solve for x:

$\mathrm{<br>}$

$3x-10+4y=8$

 

$7x-10=8$

 

$7x=18$

 

$x=\frac{18}{7}$

​ 

Now substitute x back into the first equation to find y:

$2\left(\frac{18}{7}\right)+y=5$

 

$\frac{36}{7}+y=5$

$y=\frac{35}{7}-\frac{36}{7}$

$y=-\frac{1}{7}$

So, the solution is 

$x=\frac{18}{7}$

​ and 

$y=-\frac{1}{7}$

Question 2:

System of Equations: Solve the system of linear equations:

$4x-2y=10$

 

$2x+3y=1$

 

Solution: Let's use the elimination method for this example. Multiply the first equation by 2 to make the coefficients of x in both equations the same:

$8x-4y=20$

 

$2x+3y=1$

Now subtract the second equation from the first:

$8x-4y-(2x+3y)=20-1$

 

$6x-7y=19$

  

Solve this equation for x:

$6x=7y+19$

 

$x=\frac{7}{6}y+\frac{19}{6}$

 

Substitute this expression for x back into the second equation:

$2(\frac{7}{6}y+\frac{19}{6})+3y=1$

Solve for y and then substitute it back into the expression for x.

The solution is 

$x=\frac{7}{6}y+\frac{19}{6}$

and y is a free parameter.

Question 3:

System of Equations: Solve the following system:

$3x-y=7$

 

$2x+4y=2$

Solution: Let's use the substitution method. Solve the first equation for y:

$y=3x-7$

Substitute this expression for y into the second equation:

$2x+4(3x-7)=2$

 

Simplify and solve for x:

$2x+12x-28=2$

 

$14x=30$

 

$x=\frac{15}{7}$

 ​ 

Now substitute x back into the first equation to find y:

$3\left(\frac{15}{7}\right)-y=7$

 

$45-7y=7$

 

$7y=38$

$y=\frac{38}{7}$

So, the solution is 

$x=\frac{15}{7}$

​ and 

$y=\frac{38}{7}$

These questions cover different methods for solving systems of linear equations. If you have more questions or if there's a specific aspect you'd like to explore, feel free to ask!