Math  /  Algebra

Question^ Pretest: Unit 1
Question 1 of 29 What is the solution to the system of equations below? 2x+y3z=73xy+4z=17x+2y+2z=9\begin{array}{l} 2 x+y-3 z=-7 \\ 3 x-y+4 z=-17 \\ x+2 y+2 z=9 \end{array} A. x=4,y=1,z=5x=4, y=1, z=5 B. x=4,y=4,z=3x=4, y=4, z=3 C. x=5,y=6,z=1x=-5, y=6, z=1 D. x=5,y=4,z=3x=-5, y=4, z=3

Studdy Solution

STEP 1

What is this asking? We need to find the values of xx, yy, and zz that make all three equations true at the same time! Watch out! Small arithmetic errors can mess everything up, so double-check those calculations!

STEP 2

1. Eliminate yy
2. Eliminate zz
3. Solve for xx
4. Solve for yy
5. Solve for zz

STEP 3

Let's **add** the first two equations together to eliminate yy.
We're doing this because the yy terms have opposite signs, so they'll add to zero!
2x+y3z=73xy+4z=175x+1z=24\begin{array}{rcr} 2x & +y & -3z = -7 \\ 3x & -y & +4z = -17 \\ 5x & & +1z = -24 \end{array}

STEP 4

Now, let's **multiply** the first equation by 2-2 and add it to the third equation.
We do this to make the yy coefficient match the third equation's yy coefficient, but with the opposite sign!
4x2y+6z=14x+2y+2z=93x+8z=23\begin{array}{rcr} -4x & -2y & +6z = 14 \\ x & +2y & +2z = 9 \\ -3x & & +8z = 23 \end{array}

STEP 5

We now have two new equations with xx and zz.
Let's **multiply** the first new equation by 8-8 and add it to the second new equation.
This will eliminate zz because the zz coefficients will have opposite signs and the same magnitude!
40x8z=1923x+8z=2343x=215\begin{array}{rcr} -40x & -8z & = 192 \\ -3x & +8z & = 23 \\ -43x & & = 215 \end{array}

STEP 6

**Divide** both sides of the equation 43x=215-43x = 215 by 43-43 to isolate xx.
We're dividing by 43-43 to turn the coefficient of xx into 11!
x=21543=-5x = \frac{215}{-43} = \textbf{-5}

STEP 7

**Substitute** x=5x = -5 into the equation 5x+z=245x + z = -24 (from adding the first two original equations).
We're substituting to reduce the number of unknowns in the equation!
5(5)+z=245(-5) + z = -2425+z=24-25 + z = -24

STEP 8

**Add** 2525 to both sides of the equation 25+z=24-25 + z = -24 to isolate zz.
We add 2525 to both sides to make the left side equal to zz plus zero!
z=24+25=1z = -24 + 25 = \textbf{1}

STEP 9

**Substitute** x=5x = -5 and z=1z = 1 into the first original equation 2x+y3z=72x + y - 3z = -7.
We're substituting to find yy!
2(5)+y3(1)=72(-5) + y - 3(1) = -710+y3=7-10 + y - 3 = -713+y=7-13 + y = -7

STEP 10

**Add** 1313 to both sides of the equation 13+y=7-13 + y = -7 to isolate yy.
We add 1313 to both sides to make the left side equal to yy plus zero!
y=7+13=6y = -7 + 13 = \textbf{6}

STEP 11

The solution is x=5x = -5, y=6y = 6, and z=1z = 1.
This corresponds to answer choice C!

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