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## Solving simultaneous equations

Up to now we have solved equations with only one unknown variable. When solving for two unknown variables, two equations are required and these equations are known as simultaneous equations. The solutions are the values of the unknown variables which satisfy both equations simultaneously. In general, if there are n unknown variables, then n independent equations are required to obtain a value for each of the n variables.

An example of a system of simultaneous equations is

$x+y=-13=y-2x$(1)

We have two independent equations to solve for two unknown variables. We can solve simultaneous equations algebraically using substitution and elimination methods. We will also show that a system of simultaneous equations can be solved graphically.

What are simultaneous equations

## Solving by substitution

• Use the simplest of the two given equations to express one of the variables in terms of the other.

• Substitute into the second equation. By doing this we reduce the number of equations and the number of variables by one.

• We now have one equation with one unknown variable which can be solved.

• Use the solution to substitute back into the first equation to find the value of the other unknown variable.

### Example 1: Simultaneous equations

#### Question

Solve for x and y:

$x-y=1............(1)3=y-2x.........(2)$(2)

##### Use equation $(1)$ to express x in terms of y
$x=y+1$(3)
##### Substitute x into equation $(2)$ and solve for y
$3=y-2(y+1)3=y-2y-25=-y∴y=-5$(4)
##### Substitute y back into equation $(1)$ and solve for x
$x=(-5)+1∴x=-4$(5)
##### Write the final answer
$x=-4y=-5$(6)

### Example 2: Simultaneous equations

#### Question

Solve the following system of equations:

$4y+3x=100............(1)4y-19x=12............(2)$(7)

##### Use either equation to express x in terms of y
$4y+3x=1003x=100-4yx=100-4y3$(8)
##### Substitute x into equation $(2)$ and solve for y
$4y-19100-4y3=1212y-19(100-4y)=36$(9)
$12y-1900+76y=3688y=1936∴y=22$(10)
##### Substitute y back into equation $(1)$ and solve for x
$x=100-4(22)3=100-883=123∴x=4$(11)
##### Write the final answer
$x=4y=22$(12)

## Solving by elimination

### Example 3: Simultaneous equations

#### Question

Solve the following system of equations:

$3x+y=2............(1)6x-y=25............(2)$(13)

##### Make the coefficients of one of the variables the same in both equations

The coefficients of y in the given equations are 1 and −1. Eliminate the variable y by adding equation $(1)$ and equation $(2)$ together:

$3x+y=2+6x-y=259x+0=27$(14)
##### Simplify and solve for x
$9x=27∴x=3$(15)
##### Substitute x back into either original equation and solve for y
$3(3)+y=2y=2-9∴y=-7$(16)
##### Write final answer
$x=3y=-7$(17)

### Example 4: Simultaneous equations

#### Question

Solve the following system of equations:

$2a-3b=5............(1)3a-2b=20............(2)$(18)

##### Make the coefficients of one of the variables the same in both equations

By multiplying equation $(1)$ by 3 and equation $(2)$ by 2, both coefficients of a will be 6.

$6a-9b=15-(6a-4b=40)0-5b=-25$(19)

(When subtracting two equations, be careful of the signs.)

##### Simplify and solve for b
$b=-25-5∴b=5$(20)
##### Substitute value of b back into either original equation and solve for a
$2a-3(5)=52a-15=52a=20∴a=10$(21)
##### Write final answer
$a=10b=5$(22)

## Solving graphically

Simultaneous equations can also be solved graphically. If the graphs of each linear equation are drawn, then the solution to the system of simultaneous equations is the coordinate of the point at which the two graphs intersect.

For example:

$x=2y...............(1)y=2x-3............(2)$(23)

The graphs of the two equations are shown below.

The intersection of the two graphs is $(2;1)$. So the solution to the system of simultaneous equations is $x=2$ and $y=1$.

We can also check the solution using algebraic methods.

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

$x=2y∴y=2(2y)-3$(24)

Then solve for y:

$y-4y=-3-3y=-3∴y=1$(25)

Substitute the value of y back into equation $(1)$:

$x=2(1)∴x=2$(26)

Notice that both methods give the same solution.

### Example 5: Simultaneous equations

#### Question

Solve the following system of simultaneous equations graphically:

$4y+3x=100............(1)4y-19x=12............(2)$(27)

##### Write both equations in form $y=mx+c$
$4y+3x=1004y=100-3xy=-34x+25$(28)
$4y-19x=124y=19x+12y=194x+3$(29)
##### Find the coordinates of the point of intersection

The two graphs intersect at $(4;22)$.

##### Write the final answer
$x=4y=22$(30)

### Exercise 1:

Solve for x and y:

1. $3x-14y=0$ and $x-4y+1=0$

2. $x+y=8$ and $3x+2y=21$

3. $y=2x+1$ and $x+2y+3=0$

4. $a2+b=4$ and $a4-b4=1$

5. $1x+1y=3$ and $1x-1y=11$

Solve graphically and check your answer algebraically:

1. $x+2y=1$ and $x3+y2=1$

2. $5=x+y$ and $x=y-2$

3. $3x-2y=0$ and $x-4y+1=0$

4. $x4=y2-1$ and $y4+x2=1$

5. $2x+y=5$ and $3x-2y=4$

1. $3x-14y=0$

$\therefore 3x=14y$

$\therefore x=\frac{14}{3}y$
Substitute value of x into second equation:

$x-4y+1=0$

$\frac{14}{3}y-14y+1=0$

$14y-12y+3=0$

$2y=-3$

$y=-\frac{3}{2}$
Substitute value of y back into first equation:
$\therefore x=\frac{14\left(-\frac{3}{2}\right)}{3}=-7$

2. $x=8-y$
Substitute value of x into second equation:

$3x+2y=21$

$3\left(8-y\right)+2y=21$

$24-3y+2y=21$

y=3

Substitute value of y back into first equation:
x=8-3=5

3. y=2x+1
Substitute value of y into second equation:

x+2y+3=0

x+2(2x+1)+3=0
x+4x+2+3=0
5x=-5
x=-1
Substitute value of x back into first equation:
y=2(-1)+1=-1

4. $\frac{a}{2}+b=4$
a+2b=8
a-b=4
3b=4
$b=\frac{4}{3}$
Substitute $b=\frac{4}{3}$ into the first equation:
$\frac{a}{2}+\frac{4}{3}=4$
$\frac{a}{2}=4-\frac{4}{3}$
$\frac{a}{2}=\frac{8}{3}$
$a=\frac{16}{3}$

5. $\frac{1}{x}+\frac{1}{y}=3$
$y+x=3xy$
y-x=11xy
2y=14xy
$\frac{2y}{14y}=x$
$x=\frac{1}{7}$
Substitute $x=\frac{1}{7}$ into the first equation:
$7+\frac{1}{y}=3$
$\frac{1}{y}=-4$
$y=-\frac{1}{4}$

1.

$x=-2y+1$

Substitute value of x into the second equation:

$\frac{-2y+1}{3}+\frac{y}{2}=1$
2(-2y+1) +3y=6
-4y+2+3y=6
y=-4
Substitute value of $y$ back into the first equation:
x+2(-4)=1
x-8=1
x=9

2.

5=x+y
y=5-x
Substitute value of y into second equation:
x=5-x-2
2x=3
$x=\frac{3}{2}$
Substitute value of $x$ back into first equation:
$5=\frac{3}{2}+y$

$y=\frac{10}{2}-\frac{3}{2}=\frac{7}{2}$

3.

3x-2y=0
$y=\frac{3}{2}x$
Substitute value of y into second equation:
$x-4\left(\frac{3}{2}x\right)+1=0$
x-6x+1 = 0
5x=1
$x=\frac{1}{5}$
Substitute value of x back into first equation:
$3\frac{1}{5}-2y=0$
$3-10y=0$
$10y=3$

$y=\frac{3}{10}$

4.

$\frac{x}{4}=\frac{y}{2}-1$
$x=2y-4$
$y=\frac{1}{2}x+2$
$\frac{y}{4}+\frac{x}{2}=1$
$y+2x=4$
$y=-2x+4$
Substitute $y=\frac{1}{2}x+2$ into $y=-2x+4$
$\frac{1}{2}x+2=-2x+4$
$2\frac{1}{2}x=2$
$x=\frac{4}{5}$
Substitute into $y=-2x+4$
$y=-2\frac{4}{5}+4$
$=-\frac{8}{5}+4$

$=2\frac{2}{5}$

5.

$y=-2x+5$
$y=\frac{3}{2}x-2$
$0=-\frac{7}{2}x+7$
$\frac{7}{2}x=7$
$x=2$
Substitute $x=2$ into the first equation:
$2\left(2\right)+y=5$
y=1