document.write( "Question 279308: An inventory manager orders 6 batteries, 5 cases of spark plugs, and 2 dozen pairs of wiper blades, she will pay $830. If she orders 3 batteries, 7 cases of spark plugs, and 4 dozen wiper blades, she will pay $820. If the batteries are $22 less than twice the price of a dozen wiper blades, what is the cost of each item in her order? \n" ); document.write( "

Algebra.Com's Answer #203103 by josmiceli(19441) $\"\"$ $\"About$

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Let $\"a\"$ = cost of a battery
\n" ); document.write( "Let $\"b\"$ = cost of a case of spark plugs
\n" ); document.write( "Let $\"c\"$ = cost of a dozen pair of wiper blades
\n" ); document.write( "given:
\n" ); document.write( "(1) $\"6a+%2B+5b+%2B+2c+=+830\"$
\n" ); document.write( "(2) $\"3a+%2B+7b+%2B+4c+=+820\"$
\n" ); document.write( "(3) $\"a+=+2c+-+22\"$
\n" ); document.write( "-----------
\n" ); document.write( "This is 3 equations and 3 unknowns, so it's solvable
\n" ); document.write( "Multiply both sides of (2) by $\"2\"$ and subtract
\n" ); document.write( "(1) from (2)
\n" ); document.write( "(2) $\"6a+%2B+14b+%2B+8c+=+1640\"$
\n" ); document.write( "(1) $\"-6a+-+5b+-+2c+=+830\"$
\n" ); document.write( " $\"9b+%2B+6c+=+810\"$
\n" ); document.write( " $\"9b+=+810+-+6c\"$
\n" ); document.write( " $\"b+=+90+-+%282%2F3%29%2Ac\"$
\n" ); document.write( "By substitution:
\n" ); document.write( "(2) $\"3a+%2B+7b+%2B+4c+=+820\"$
\n" ); document.write( " $\"3%2A%282c+-+22%29+%2B+7%2A%2890+-+%282%2F3%29%2Ac%29+%2B+4c+=+820\"$
\n" ); document.write( " $\"6c+-+66+%2B+630+-+%2814%2F3%29%2Ac+%2B+4c+=+820\"$
\n" ); document.write( " $\"%2816%2F3%29%2Ac+=+820+-+630+%2B+66\"$
\n" ); document.write( " $\"%2816%2F3%29%2Ac+=+256\"$
\n" ); document.write( " $\"16c+=+768\"$
\n" ); document.write( " $\"c+=+48\"$
\n" ); document.write( "And, from (3),
\n" ); document.write( " $\"a+=+2c+-+22\"$
\n" ); document.write( " $\"a+=+2%2A48+-+22\"$
\n" ); document.write( " $\"a+=+74\"$
\n" ); document.write( "And, from (1)
\n" ); document.write( "(1) $\"6a+%2B+5b+%2B+2c+=+830\"$
\n" ); document.write( " $\"6%2A74+%2B+5b+%2B+2%2A48+=+830\"$
\n" ); document.write( " $\"444+%2B+5b+%2B+96++=+830\"$
\n" ); document.write( " $\"5b+=+830+-+444+-+96\"$
\n" ); document.write( " $\"5b+=+290\"$
\n" ); document.write( " $\"b+=+58\"$
\n" ); document.write( "$74 = cost of a battery
\n" ); document.write( "$58 = cost of a case of spark plugs
\n" ); document.write( "$48 = cost of a dozen pair of wiper blades
\n" ); document.write( "---------------------------------
\n" ); document.write( "check answer:
\n" ); document.write( "(2) $\"3a+%2B+7b+%2B+4c+=+820\"$
\n" ); document.write( " $\"3%2A74+%2B+7%2A58+%2B+4%2A48+=+820\"$
\n" ); document.write( " $\"222+%2B+406+%2B+192+=+820\"$
\n" ); document.write( " $\"820+=+820\"$
\n" ); document.write( "OK \n" ); document.write( "

\n" );