To eliminate the \( y \)-terms and solve for \( x \) in the fewest steps, let’s consider the given equations:
First Equation:
[tex]\[ 5x - 4y = 28 \][/tex]
Second Equation:
[tex]\[ 3x - 9y = 30 \][/tex]
We want to eliminate the \( y \)-terms by making their coefficients equal in magnitude but opposite in sign.
To find the constants by which the equations should be multiplied, we need to focus on the coefficients of \( y \) in each equation: -4 and -9. We seek the Least Common Multiple (LCM) of these coefficients in absolute terms, which is 36.
Next, we adjust the coefficients of \( y \) such that they become opposites:
1. Multiply the first equation ( \( 5x - 4y = 28 \) ) by 9 so that the coefficient of \( y \) becomes \( -36 \):
[tex]\[ 9 \times (5x - 4y) = 9 \times 28 \][/tex]
[tex]\[ 45x - 36y = 252 \][/tex]
2. Multiply the second equation ( \( 3x - 9y = 30 \) ) by 4 so that the coefficient of \( y \) becomes \( -36 \):
[tex]\[ 4 \times (3x - 9y) = 4 \times 30 \][/tex]
[tex]\[ 12x - 36y = 120 \][/tex]
Thus, the coefficients of \( y \) in both equations are \( -36 \), meaning they are now equal in magnitude.
Therefore, the first equation should be multiplied by \( 9 \) and the second equation should be multiplied by \( 4 \) before adding the equations together to eliminate the \( y \)-term.
The correct multipliers are [tex]\( \boxed{9 \text{ and } 4} \)[/tex].
