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Math  /  Algebra

QuestionRework problem 18 from section 6.1 of your text, involving two constants, cc and dd. Use the following matrix equation instead of that found in your text to find the values of cc and dd which make the equation true. c[−9−5]+d[21]=[−114−63]c\left[\begin{array}{l} -9 \\ -5 \end{array}\right]+d\left[\begin{array}{l} 2 \\ 1 \end{array}\right]=\left[\begin{array}{l} -114 \\ -63 \end{array}\right] (1) To make this equation true, set c=c= □\square (2) And set d=d= □\square Submit answer Next item

Studdy Solution

STEP 1

1. We are given a vector equation involving two constants, c c and d d .
2. The equation is a linear combination of two vectors that equals a resultant vector.
3. We need to solve for the values of c c and d d that satisfy the equation.

STEP 2

1. Set up a system of linear equations from the given matrix equation.
2. Solve the system of equations for c c and d d .

STEP 3

First, express the given matrix equation as a system of linear equations. The equation is:
c[−9−5]+d[21]=[−114−63] c\left[\begin{array}{l} -9 \\ -5 \end{array}\right] + d\left[\begin{array}{l} 2 \\ 1 \end{array}\right] = \left[\begin{array}{l} -114 \\ -63 \end{array}\right]
This can be written as:
1. −9c+2d=−114 -9c + 2d = -114
2. −5c+1d=−63 -5c + 1d = -63

STEP 4

Solve the system of equations using the method of substitution or elimination. We'll use elimination here.
First, multiply the second equation by 2 to align the coefficients of d d :
2(−5c+d)=2(−63) 2(-5c + d) = 2(-63) −10c+2d=−126 -10c + 2d = -126
Now we have:
1. −9c+2d=−114 -9c + 2d = -114
2. −10c+2d=−126 -10c + 2d = -126
Subtract the first equation from the second equation to eliminate d d :
(−10c+2d)−(−9c+2d)=−126−(−114) (-10c + 2d) - (-9c + 2d) = -126 - (-114) −10c+2d+9c−2d=−126+114 -10c + 2d + 9c - 2d = -126 + 114 −c=−12 -c = -12
Solve for c c :
c=12 c = 12

STEP 5

Substitute c=12 c = 12 back into one of the original equations to solve for d d . We'll use the second equation:
−5c+d=−63 -5c + d = -63 −5(12)+d=−63 -5(12) + d = -63 −60+d=−63 -60 + d = -63
Add 60 to both sides to solve for d d :
d=−63+60 d = -63 + 60 d=−3 d = -3
The values of c c and d d that make the equation true are:
(1) To make this equation true, set c=12 c = 12
(2) And set d=−3 d = -3

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