3 Variable Systems of Equations – Elimination Method

Topics

Uncategorized

Try VividMath Premium to unlock full access

Start Free Trial

Lets see your results!

Points

Score

Time

Retry Quiz

Answered
Review

Question 1 of 5

Solve the following simultaneous equations by elimination.

$3x-y+2z=5$

$4x+2y-z=6$

$5x-3y+z=1$

$x=$ (1)

$y=$ (2)

$z=$ (2)

Incorrect

Help Video

Elimination Method

$1)$ make sure 2 equations have a variable with same coefficients but opposite signs
$2)$ add or subtract the equations so that one variable is cancelled
$3)$ solve for the variable that remains
$4)$ substitute known value to one of the equations to solve for the other variable

First, label the equations

$1$ ,

$2$ and

$3$ respectively.

$3x-y+2z$	$=$	$5$	Equation $1$
$4x+2y-z$	$=$	$6$	Equation $2$
$5x-3y+z$	$=$	$1$	Equation $3$

Next, add equation

$2$ to equation

$3$ since their variable

$z$ both have

$1$ as coefficient but with opposite signs. Label the result as equation

$A$ .

$4x+2y-z$	$=$	$6$
$+$ $5x-3y+z$	$=$	$1$	$-z+z$ cancels out

$9x-y$	$=$	$7$	Equation $A$

Next, multiply the values of equation

$2$ by

$2$ and add the product to equation

$1$ . Label the result as equation

$B$ .

$2xx$ $(4x+2y-z)$	$=$	$2xx$ $6$	Multiply equation $2$ by $2$
$8x+4y-2z$	$=$	$12$

$8x+4y-2z$	$=$	$12$
$+$ $3x-y+2z$	$=$	$5$	$+2z-2z$ cancels out

$11x+3y$	$=$	$17$	Equation $B$

Next, multiply the values of equation

$A$ by

$3$ and add the product to equation

$B$ to find the value of $x$ .

$3xx$ $(9x-y)$	$=$	$3xx$ $7$	Multiply equation $A$ by $3$
$27x-3y$	$=$	$21$

$27x-3y$	$=$	$21$
$+$ $11x+3y$	$=$	$17$

$38x$	$=$	$38$	$-3y+3y$ cancels out
$38x$ $div38$	$=$	$38$ $div38$	Divide both sides by $38$
$x$	$=$	$1$

Now, substitute the value of $x$ into equation

$A$ to solve for $y$ .

$9$ $x$ $-y$	$=$	$7$	Equation $A$
$9($ $1$ $)-y$	$=$	$7$	$x=1$
$9-y$	$=$	$7$
$9-y$ $-9$	$=$	$7$ $-9$	Subtract $9$ from both sides
$-y$	$=$	$-2$
$-y$ $times(-1)$	$=$	$-2$ $times(-1)$	Multiply both sides by $-1$
$y$	$=$	$2$

Finally, substitute the value of $x$ and $y$ into equation

$1$ to solve for $z$ .

$3$ $x$ $-$ $y$ $+2z$	$=$	$5$	Equation $1$
$3($ $1$ $)-$ $2$ $+2z$	$=$	$5$	$x=1$ and $y=2$
$3-2+2z$	$=$	$5$
$1+2z$	$=$	$5$
$1+2z$ $-1$	$=$	$5$ $-1$	Subtract $1$ from both sides
$2z$	$=$	$4$
$2z$ $div2$	$=$	$4$ $div2$	Divide both sides by $2$
$z$	$=$	$2$

$x=1$

$y =2$

$z =2$

Question 2 of 5

Solve the following simultaneous equations by elimination.

$3x+5y+z=5$

$9x+2y+7z=32$

$6x+3y+4z=19$

$x=$ (3)

$y=$ (-1)

$z=$ (1)

Incorrect

Help Video

Elimination Method

$1)$ make sure 2 equations have a variable with same coefficients but opposite signs
$2)$ add or subtract the equations so that one variable is cancelled
$3)$ solve for the variable that remains
$4)$ substitute known value to one of the equations to solve for the other variable

First, label the equations

$1$ ,

$2$ and

$3$ respectively.

$3x+5y+z$	$=$	$5$	Equation $1$
$9x+2y+7z$	$=$	$32$	Equation $2$
$6x+3y+4z$	$=$	$19$	Equation $3$

Next, multiply the values of equation

$1$ by

$3$ and subtract equation

$1$ from the product. Label the result as equation

$A$ .

$3xx$ $(3x+5y+z)$	$=$	$3xx$ $5$	Multiply equation $1$ by $3$
$9x+15y+3z$	$=$	$15$

$9x+15y+3z$	$=$	$15$
$-$ $9x-2y-7z$	$=$	$32$	$9x-9x$ cancels out

$13y-4z$	$=$	$-17$	Equation $A$

Then, multiply the values of equation

$1$ by

$2$ and subtract equation

$3$ from the product. Label the result as equation

$B$ .

$2xx$ $(3x+5y+z)$	$=$	$2xx$ $5$	Multiply equation $1$ by $2$
$6x+10y+2z$	$=$	$10$

$6x+10y+2z$	$=$	$10$
$-$ $6x-3y-4z$	$=$	$19$	$6x-6x$ cancels out

$7y-2z$	$=$	$-9$	Equation $B$

Now, multiply the values of equation

$B$ by

$2$ and subtract equation

$A$ from the product to find the value of $y$ .

$2xx$ $(7y-2z)$	$=$	$2xx$ $-9$	Multiply equation $B$ by $2$
$14y-4z$	$=$	$-18$

$14y-4z$	$=$	$-18$
$-$ $13y+4z$	$=$	$17$	$-4z+4z$ cancels out

$y$	$=$	$-1$

Now, substitute the value of $y$ into equation

$B$ to solve for $z$ .

$7$ $y$ $-2z$	$=$	$-9$	Equation $B$
$7($ $-1$ $)-2z$	$=$	$-9$	$y=-1$
$-7-2z$	$=$	$-9$
$-7-2z$ $+7$	$=$	$-9$ $+7$	Add $7$ to both sides
$-2z$	$=$	$-2$
$-2z$ $div(-2)$	$=$	$-2$ $div(-2)$	Divide both sides by $-2$
$z$	$=$	$1$

Finally, substitute the value of $y$ and $z$ into equation

$1$ to solve for $x$ .

$3x+5$ $y$ $+$ $z$	$=$	$5$	Equation $1$
$3x+5($ $-1$ $)+$ $1$	$=$	$5$	$y=-1$ and $z=1$
$3x-5+1$	$=$	$5$
$3x-4$	$=$	$5$
$3x-4$ $+4$	$=$	$5$ $+4$	Add $4$ to both sides
$3x$	$=$	$9$
$3x$ $div3$	$=$	$9$ $div3$	Divide both sides by $3$
$x$	$=$	$3$

$x=3$

$y =-1$

$z =1$

Question 3 of 5

Solve the following simultaneous equations by elimination.

$x+2y+5z=2$

$3x+4y-4z=21$

$2x-y+z=3$

$x=$ (3)

$y=$ (2)

$z=$ (-1)

Incorrect

Help Video

Elimination Method

$1)$ make sure 2 equations have a variable with same coefficients but opposite signs
$2)$ add or subtract the equations so that one variable is cancelled
$3)$ solve for the variable that remains
$4)$ substitute known value to one of the equations to solve for the other variable

First, label the equations

$1$ ,

$2$ and

$3$ respectively.

$x+2y+5z$	$=$	$2$	Equation $1$
$3x+4y-4z$	$=$	$21$	Equation $2$
$2x-y+z$	$=$	$3$	Equation $3$

Next, multiply the values of equation

$1$ by

$3$ and subtract equation

$2$ from the product. Label the result as equation

$A$ .

$3xx$ $(x+2y+5z)$	$=$	$3xx$ $2$	Multiply equation $1$ by $3$
$3x+6y+15z$	$=$	$6$

$3x+6y+15z$	$=$	$6$
$-$ $3x-4y+4z$	$=$	$21$	$3x-3x$ cancels out

$2y+19z$	$=$	$-15$	Equation $A$

Then, multiply the values of equation

$1$ by

$2$ and subtract equation

$3$ from the product. Label the result as equation

$B$ .

$2xx$ $(x+2y+5z)$	$=$	$2xx$ $2$	Multiply equation $1$ by $2$
$2x+4y+10z$	$=$	$4$

$2x+4y+10z$	$=$	$4$
$-$ $2x+y-z$	$=$	$-3$	$2x-2x$ cancels out

$5y+9z$	$=$	$1$	Equation $B$

Now, multiply the values of equation

$A$ by

$5$ and the values of equation

$B$ by

$2$ , then get their difference to find the value of $z$ .

$5xx$ $(2y+19z)$	$=$	$5xx$ $-15$	Multiply equation $A$ by $5$
$10y+95z$	$=$	$-75$

$2xx$ $(5y+9z)$	$=$	$2xx$ $1$	Multiply equation $B$ by $2$
$10y+18z$	$=$	$2$

$10y+18z$	$=$	$2$
$-$ $10y-95z$	$=$	$75$	$10y-10y$ cancels out

$-77z$	$=$	$77$
$-77z$ $div(-77)$	$=$	$77$ $div(-77)$	Divide both sides by $-77$
$z$	$=$	$-1$

Now, substitute the value of $z$ into equation

$B$ to solve for $y$ .

$5y+9$ $z$	$=$	$1$	Equation $B$
$5y+9($ $-1$ $)$	$=$	$1$	$z=-1$
$5y-9$	$=$	$1$
$5y-9$ $+9$	$=$	$1$ $+9$	Add $9$ to both sides
$5y$	$=$	$10$
$5y$ $div5$	$=$	$10$ $div5$	Divide both sides by $5$
$y$	$=$	$2$

Finally, substitute the value of $y$ and $z$ into equation

$1$ to solve for $x$ .

$x+2$ $y$ $+5$ $z$	$=$	$2$	Equation $1$
$x+2($ $2$ $)+5($ $-1$ $)$	$=$	$2$	$y=2$ and $z=-1$
$x+4-5$	$=$	$2$
$x-1$	$=$	$2$
$x-1$ $+1$	$=$	$2$ $+1$	Add $1$ to both sides
$x$	$=$	$3$

$x=3$

$y =2$

$z =-1$

Question 4 of 5

Solve the following simultaneous equations by elimination.

$-2x-y+5z=5$

$3x+2y-z=7$

$-4x+4y+2z=12$

$x=$ (1)

$y=$ (3)

$z=$ (2)

Incorrect

Help Video

Elimination Method

$1)$ make sure 2 equations have a variable with same coefficients but opposite signs
$2)$ add or subtract the equations so that one variable is cancelled
$3)$ solve for the variable that remains
$4)$ substitute known value to one of the equations to solve for the other variable

First, label the equations

$1$ ,

$2$ and

$3$ respectively.

$-2x-y+5z$	$=$	$5$	Equation $1$
$3x+2y-z$	$=$	$7$	Equation $2$
$-4x+4y+2z$	$=$	$12$	Equation $3$

Next, multiply the values of equation

$1$ by

$2$ and add equation

$2$ to the product. Label the result as equation

$A$ .

$2xx$ $(-2x-y+5z)$	$=$	$2xx$ $5$	Multiply equation $1$ by $2$
$-4x-2y+10z$	$=$	$10$

$-4x-2y+10z$	$=$	$10$
$+$ $3x+2y-z$	$=$	$7$	$-2y+2y$ cancels out

$-x+9z$	$=$	$17$	Equation $A$

Then, multiply the values of equation

$2$ by

$2$ and subtract equation

$3$ from the product. Label the result as equation

$B$ .

$2xx$ $(3x+2y-z)$	$=$	$2xx$ $7$	Multiply equation $2$ by $2$
$6x+4y-2z$	$=$	$14$

$6x+4y-2z$	$=$	$14$
$-$ $4x-4y-2z$	$=$	$-12$	$4y-4y$ cancels out

$10x-4z$	$=$	$2$	Equation $B$

Now, multiply the values of equation

$A$ by

$10$ and add the product to equation

$A$ to find the value of $z$ .

$10xx$ $(-x+9z)$	$=$	$10xx$ $17$	Multiply equation $A$ by $10$
$-10x+90z$	$=$	$170$

$10x-4z$	$=$	$2$
$+$ $-10x+90z$	$=$	$170$	$10x-10x$ cancels out

$86z$	$=$	$172$
$86z$ $div86$	$=$	$172$ $div86$	Divide both sides by $86$
$z$	$=$	$2$

Now, substitute the value of $z$ into equation

$B$ to solve for $x$ .

$10x-4$ $z$	$=$	$2$	Equation $B$
$10x-4($ $2$ $)$	$=$	$2$	$y=-1$
$10x-8$	$=$	$2$
$10x-8$ $+8$	$=$	$2$ $+8$	Add $8$ to both sides
$10x$	$=$	$10$
$10x$ $div10$	$=$	$10$ $div10$	Divide both sides by $10$
$x$	$=$	$1$

Finally, substitute the value of $x$ and $z$ into equation

$2$ to solve for $y$ .

$3$ $x$ $+2y-$ $z$	$=$	$7$	Equation $2$
$3($ $1$ $)+2y-$ $2$	$=$	$7$	$x=1$ and $z=2$
$3+2y-2$	$=$	$7$
$1+2y$	$=$	$7$
$1+2y$ $-1$	$=$	$7$ $-1$	Subtract $1$ from both sides
$2y$	$=$	$6$
$2y$ $div2$	$=$	$6$ $div2$	Divide both sides by $2$
$y$	$=$	$3$

$x=1$

$y =3$

$z =2$

Question 5 of 5

Solve the following simultaneous equations by elimination.

$2a-b-4c=1$

$a+6b-c=8$

$3a+4b-2c=11$

$a=$ (3)

$b=$ (1)

$c=$ (1)

Incorrect

Help Video

Elimination Method

$1)$ make sure 2 equations have a variable with same coefficients but opposite signs
$2)$ add or subtract the equations so that one variable is cancelled
$3)$ solve for the variable that remains
$4)$ substitute known value to one of the equations to solve for the other variable

First, label the equations

$1$ ,

$2$ and

$3$ respectively.

$2a-b-4c$	$=$	$1$	Equation $1$
$a+6b-c$	$=$	$8$	Equation $2$
$3a+4b-2c$	$=$	$11$	Equation $3$

Next, multiply the values of equation

$2$ by

$2$ and subtract equation

$1$ from the product. Label the result as equation

$A$ .

$2xx$ $(a+6b-c)$	$=$	$2xx$ $8$	Multiply equation $2$ by $2$
$2a+12b-2c$	$=$	$16$

$2a+12b-2c$	$=$	$16$
$-$ $2a+b+4c$	$=$	$1$	$2a-2a$ cancels out

$13b+2c$	$=$	$15$	Equation $A$

Then, multiply the values of equation

$2$ by

$3$ and subtract equation

$3$ from the product. Label the result as equation

$B$ .

$3xx$ $(a+6b-c)$	$=$	$3xx$ $8$	Multiply equation $2$ by $3$
$3a+18b-3c$	$=$	$24$

$3a+18b-3c$	$=$	$24$
$-$ $3a-4b+2c$	$=$	$-11$	$6x-6x$ cancels out

$14b-c$	$=$	$13$	Equation $B$

Now, multiply the values of equation

$B$ by

$2$ and add equation

$A$ to the product to find the value of $y$ .

$2xx$ $(14b-c)$	$=$	$2xx$ $13$	Multiply equation $B$ by $2$
$28b-2c$	$=$	$26$

$28b-2c$	$=$	$26$
$+$ $13b+2c$	$=$	$15$	$-4z+4z$ cancels out

$41b$	$=$	$41$
$41b$ $div41$	$=$	$41$ $div41$	Divide both sides by $41$
$b$	$=$	$1$

Now, substitute the value of $b$ into equation

$A$ to solve for $c$ .

$13$ $b$ $+2c$	$=$	$15$	Equation $A$
$13($ $1$ $)+2c$	$=$	$15$	$y=1$
$13+2c$	$=$	$15$
$13+2c$ $-13$	$=$	$15$ $-13$	Subtract $13$ from both sides
$2c$	$=$	$2$
$2c$ $div2$	$=$	$2$ $div2$	Divide both sides by $2$
$c$	$=$	$1$

Finally, substitute the value of $b$ and $c$ into equation

$2$ to solve for $a$ .

$a+6$ $b$ $-$ $c$	$=$	$8$	Equation $1$
$a+6($ $1$ $)-$ $1$	$=$	$8$	$b=1$ and $c=1$
$a+6-1$	$=$	$8$
$a+5$	$=$	$8$
$a+5$ $-5$	$=$	$8$ $-5$	Subtract $5$ from both sides
$a$	$=$	$3$

$a=3$

$b =1$

$c =1$

3 Variable Systems of Equations – Elimination Method

Try VividMath Premium to unlock full access

Quiz summary

Information

1. Question

Hint

Nice Job!

Elimination Method

2. Question

Hint

Keep Going!

Elimination Method

3. Question

Hint

Excellent!

Elimination Method

4. Question

Hint

Well Done!

Elimination Method

5. Question

Hint

Great Work!

Elimination Method

Quizzes