Substitution Method 4

Question 1 of 4

Solve the following simultaneous equations by substitution.

b = 2 a + 2

$b=2a+2$

2 a + 5 b = 10

$2a+5b=10$

$a =$ $a=$ (0)

$b =$ $b=$ (2)

Question 2 of 4

Solve the following simultaneous equations by substitution.

2 m + 5 n = 5

$2m+5n=5$

m - 5 n = 10

$m-5n=10$

$m =$ $m=$ (5)

$n =$ $n=$ (-1)

Question 3 of 4

Solve the following simultaneous equations by substitution.

4 x - 5 y = 14

$4x-5y=14$

2 x + 3 y = - 4

$2x+3y=-4$

$x =$ $x=$ (1)

$y =$ $y=$ (-2)

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Substitution Method

$1)$ $1)$ make one variable the subject
$2)$ $2)$ substitute into second equation
$3)$ $3)$ solve the second equation
$4)$ $4)$ substitute back

First, label the two equations

1

$1$ and

2

$2$ respectively.

$4 x - 5 y$ $4x-5y$	$=$ $=$	$14$ $14$	Equation $1$ $1$
$2 x + 3 y$ $2x+3y$	$=$ $=$	$- 4$ $-4$	Equation $2$ $2$

Next, solve for $x$ $x$ in Equation

2

$2$ .

$2 x + 3 y$ $2x+3y$	$=$ $=$	$- 4$ $-4$
$2 x + 3 y$ $2x+3y$ $- 3 y$ $-3y$	$=$ $=$	$- 4$ $-4$ $- 3 y$ $-3y$	Subtract $3 y$ $3y$ from both sides
$2 x$ $2x$ $\div 2$ $div2$	$=$ $=$	$(- 4 - 3 y)$ $(-4-3y)$ $\div 2$ $div2$	Divide both sides by $2$ $2$

$x$ $x$	$=$ $=$	$\frac{- 4 - 3 y}{2}$ $(-4-3y)/2$	Simplify

Substitute $x$ $x$ into Equation

1

$1$ .

$4$ $4$ $x$ $x$ $- 5 y$ $-5y$	$=$ $=$	$14$ $14$	Equation $1$ $1$

$4$ $4$ $(\frac{- 4 - 3 y}{2})$ $((-4-3y)/2)$ $- 5 y$ $-5y$	$=$ $=$	$14$ $14$	$x = \frac{- 4 - 3 y}{2}$ $x=(-4-3y)/2$

$\left(4\left(\frac{-4-3y}{2}\right)\color{#CC0000}{\times2}\right)-(5y\color{#CC0000}{\times2})$	$=$ $=$	$14$ $14$ $\times 2$ $times2$	Multiply all values by $2$ $2$

$4 (- 4 - 3 y) - 10 y$ $4(-4-3y)-10y$	$=$ $=$	$28$ $28$
$- 16 - 12 y - 10 y$ $-16-12y-10y$	$=$ $=$	$28$ $28$	Distribute $4$ $4$ inside the parenthesis
$- 16 - 22 y$ $-16-22y$	$=$ $=$	$28$ $28$	Simplify
$- 16 - 22 y$ $-16-22y$ $+ 16$ $+16$	$=$ $=$	$28$ $28$ $+ 16$ $+16$	Solve for $y$ $y$
$- 22 y$ $-22y$	$=$ $=$	$44$ $44$
$- 22 y$ $-22y$ $\div - 22$ $div-22$	$=$ $=$	$44$ $44$ $\div - 22$ $div-22$	Divide both sides by $- 22$ $-22$
$y$ $y$	$=$ $=$	$- 2$ $-2$

Now, substitute the value of $y$ $y$ into Equation

2

$2$

$2 x + 3$ $2x+3$ $y$ $y$	$=$ $=$	$- 4$ $-4$	Equation $2$ $2$
$2 x + 3$ $2x+3$ $(- 2)$ $(-2)$	$=$ $=$	$- 4$ $-4$	$y = - 2$ $y=-2$
$2 x - 6$ $2x-6$ $+ 6$ $+6$	$=$ $=$	$- 4$ $-4$ $+ 6$ $+6$	Add $6$ $6$ to both sides
$2 x$ $2x$ $\div 2$ $div2$	$=$ $=$	$2$ $2$ $\div 2$ $div2$	Divide both sides by $2$ $2$
$x$ $x$	$=$ $=$	$1$ $1$

x = 1, y = - 2

$x=1, y=-2$

Question 4 of 4

Solve the following simultaneous equations by substitution.

7 x - 2 y = 6

$7x-2y=6$

3 x + 4 y = 22

$3x+4y=22$

$x =$ $x=$ (2)

$y =$ $y=$ (4)

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Substitution Method

$1)$ $1)$ make one variable the subject
$2)$ $2)$ substitute into second equation
$3)$ $3)$ solve the second equation
$4)$ $4)$ substitute back

First, label the two equations

1

$1$ and

2

$2$ respectively.

$7 x - 2 y$ $7x-2y$	$=$ $=$	$6$ $6$	Equation $1$ $1$
$3 x + 4 y$ $3x+4y$	$=$ $=$	$22$ $22$	Equation $2$ $2$

Next, solve for $y$ $y$ in Equation

1

$1$ .

$7 x - 2 y$ $7x-2y$	$=$ $=$	$6$ $6$
$7 x - 2 y$ $7x-2y$ $+ 2 y$ $+2y$	$=$ $=$	$6$ $6$ $+ 2 y$ $+2y$	Add $2 y$ $2y$ to both sides
$7 x$ $7x$ $- 6$ $-6$	$=$ $=$	$6 + 2 y$ $6+2y$ $- 6$ $-6$	Subtract $2 y$ $2y$ from both sides
$(7 x - 6)$ $(7x-6)$ $\div 2$ $div2$	$=$ $=$	$2 y$ $2y$ $\div 2$ $div2$	Divide both sides by $2$ $2$

$\frac{7 x - 6}{2}$ $(7x-6)/2$	$=$ $=$	$y$ $y$	Simplify

$y$ $y$	$=$ $=$	$\frac{7 x - 6}{2}$ $(7x-6)/2$	Simplify

Substitute $y$ $y$ into Equation

2

$2$ .

$3 x + 4$ $3x+4$ $y$ $y$	$=$ $=$	$22$ $22$	Equation $2$ $2$

$3 x + 4$ $3x+4$ $(\frac{7 x - 6}{2})$ $((7x-6)/2)$	$=$ $=$	$22$ $22$	$y = \frac{7 x - 6}{2}$ $y=(7x-6)/2$

$(3x\color{#CC0000}{\times2})+\left(4\left(\frac{7x-6}{2}\right)\color{#CC0000}{\times2}\right)$	$=$ $=$	$22$ $22$ $\times 2$ $times2$	Multiply all values by $2$ $2$

$6 x + 4 (7 x - 6)$ $6x+4(7x-6)$	$=$ $=$	$44$
$6x+28x-24$	$=$	$44$	Distribute $4$ inside the parenthesis
$34x-24$	$=$	$44$	Simplify
$34x-24$ $+24$	$=$	$44$ $+24$	Solve for $x$
$34x$	$=$	$68$
$34x$ $div34$	$=$	$68$ $div34$	Divide both sides by $34$
$x$	$=$	$2$

Now, substitute the value of $x$ into Equation

$1$

$7$ $x$ $-2y$	$=$	$6$	Equation $1$
$7$ $(2)$ $-2y$	$=$	$6$	$x=2$
$14-2y$ $-14$	$=$	$6$ $-14$	Subtract $14$ from both sides
$-2y$ $div-2$	$=$	$-8$ $div-2$	Divide both sides by $-2$
$y$	$=$	$4$

$x=2, y=4$

$2 a + 5$ $2a+5$ $b$ $b$	$=$ $=$	$10$ $10$	Equation $2$ $2$
$2 a + 5$ $2a+5$ $(2 a + 2)$ $(2a+2)$	$=$ $=$	$10$ $10$	$b = 2 a + 2$ $b=2a+2$
$2 a + 10 a + 10$ $2a+10a+10$	$=$ $=$	$10$ $10$	Distribute $5$ $5$ inside the parenthesis
$12 a + 10$ $12a+10$	$=$ $=$	$10$ $10$	Simplify
$12 a + 10$ $12a+10$ $- 10$ $-10$	$=$ $=$	$10$ $10$ $- 10$ $-10$	Solve for $a$ $a$
$12 a$ $12a$	$=$ $=$	$0$ $0$
$12 a$ $12a$ $\div 12$ $div12$	$=$ $=$	$0$ $0$ $\div 12$ $div12$	Divide both sides by $12$ $12$
$a$ $a$	$=$ $=$	$0$ $0$

$b$ $b$	$=$ $=$	$2$ $2$ $a$ $a$ $+ 2$ $+2$	Equation $1$ $1$
$b$ $b$	$=$ $=$	$2$ $2$ $(0)$ $(0)$ $+ 2$ $+2$	$a = 0$ $a=0$
$b$ $b$	$=$ $=$	$0 + 2$ $0+2$
$b$ $b$	$=$ $=$	$2$ $2$

$m - 5 n$ $m-5n$	$=$ $=$	$10$ $10$
$m - 5 n$ $m-5n$ $+ 5 n$ $+5n$	$=$ $=$	$10$ $10$ $+ 5 n$ $+5n$	Add $5 n$ $5n$ to both sides
$m$ $m$	$=$ $=$	$10 + 5 n$ $10+5n$	Simplify

$2$ $2$ $m$ $m$ $+ 5 n$ $+5n$	$=$ $=$	$5$ $5$	Equation $1$ $1$
$2$ $2$ $(10 + 5 n)$ $(10+5n)$ $+ 5 n$ $+5n$	$=$ $=$	$5$ $5$	$m = 10 + 5 n$ $m=10+5n$
$20 + 10 n + 5 n$ $20+10n+5n$	$=$ $=$	$5$ $5$	Distribute $2$ $2$ inside the parenthesis
$20 + 15 n$ $20+15n$	$=$ $=$	$5$ $5$	Simplify
$20 + 15 n$ $20+15n$ $- 20$ $-20$	$=$ $=$	$5$ $5$ $- 20$ $-20$	Solve for $n$ $n$
$15 n$ $15n$	$=$ $=$	$- 15$ $-15$
$15 n$ $15n$ $\div 15$ $div15$	$=$ $=$	$- 15$ $-15$ $\div 15$ $div15$	Divide both sides by $15$ $15$
$n$ $n$	$=$ $=$	$- 1$ $-1$

$m - 5$ $m-5$ $n$ $n$	$=$ $=$	$10$ $10$	Equation $1$ $1$
$m - 5$ $m-5$ $(- 1)$ $(-1)$	$=$ $=$	$10$ $10$	$n = - 1$ $n=-1$
$m + 5$ $m+5$ $- 5$ $-5$	$=$ $=$	$10$ $10$ $- 5$ $-5$	Subtract $5$ $5$ from both sides
$m$ $m$	$=$ $=$	$5$ $5$

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Quiz summary

Information

1. Question

Hint

Well Done!

Substitution Method

2. Question

Hint

Fantastic!

Substitution Method

3. Question

Hint

Great Work!

Substitution Method

4. Question

Hint

Well Done!

Substitution Method

Quizzes

$b$ $b$	$=$ $=$	$2 a + 2$ $2a+2$	Equation $1$ $1$
$2 a + 5 b$ $2a+5b$	$=$ $=$	$10$ $10$	Equation $2$ $2$

$2 m + 5 n$ $2m+5n$	$=$ $=$	$5$ $5$	Equation $1$ $1$
$m - 5 n$ $m-5n$	$=$ $=$	$10$ $10$	Equation $2$ $2$