Full solution

Q. Solve by elimination

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\begin{array} -7 y+5 x-11=0 \\ 3 y+20 x-44=0 \end{array}.

Write Equations: First, let's write down the system of equations to be solved by elimination: $\newline$ $\begin{align*} -7y + 5x - 11 &= 0 \\ 3y + 20x - 44 &= 0 \end{align*}$ $\newline$ We want to eliminate one of the variables by combining the equations. To do this, we need to make the coefficients of one of the variables (either x or y) the same or opposites in both equations.
Multiply Equations: Let's choose to eliminate the y variable. To do this, we need to multiply the first equation by $3$ and the second equation by $7$ so that the coefficients of y in both equations are opposites. $\newline$ $\begin{align*} 3(-7y + 5x - 11) &= 3(0) \\ 7(3y + 20x - 44) &= 7(0) \end{align*}$
Perform Multiplication: Now, let's perform the multiplication: $\newline$ $\begin{align*} -21y + 15x - 33 &= 0 \\ 21y + 140x - 308 &= 0 \end{align*}$
Eliminate y: Next, we add the two equations together to eliminate y: $\newline$ $(-21y + 15x - 33) + (21y + 140x - 308) = 0 + 0$
Add Equations: After adding the equations, we get: $\newline$ $15x + 140x - 33 - 308 = 0$
Combine Terms: Combining like terms gives us: $\newline$ $155x - 341 = 0$
Solve for x: Now, we solve for x by adding $341$ to both sides of the equation: $\newline$ $155x = 341$
Divide by $155$ : Next, we divide both sides by $155$ to find the value of x: $\newline$ $x = \frac{341}{155}$
Substitute into Equation: Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the first equation: $\newline$ $-7y + 5\left(\frac{341}{155}\right) - 11 = 0$
Multiply $5$ : We multiply $5$ by $\frac{341}{155}$ to get: $\newline$ $-7y + \frac{1705}{155} - 11 = 0$
Convert $11$ : Next, we convert $11$ to a fraction with the same denominator to combine it with $\frac{1705}{155}$ : $\newline$ $-7y + \frac{1705}{155} - \frac{1705}{155} = 0$