Questions: Solve the following system of equations with the substitution method: x-4y=21 3x-5y=35 Answer: (x, y)= Preview x : Preview y : Enter your answers as integers or as reduced fractions in the form A/B.

$Solve the following system of equations with the substitution method: x-4y=21 3x-5y=35 Answer: (x, y)= Preview x : Preview y : Enter your answers as integers or as reduced fractions in the form A/B.$

Transcript text: Solve the following system of equations with the substitution method: \[ \left\{\begin{array}{l} x-4 y=21 \\ 3 x-5 y=35 \end{array}\right. \] Answer: $(x, y)=$ $\square$ $\square$ Preview $x$ : Preview $y$ : Enter your answers as integers or as reduced fractions in the form A/B.

Solution

Solution Steps

To solve the system of equations using the substitution method, we first solve one of the equations for one variable in terms of the other. Then, substitute this expression into the other equation to find the value of one variable. Finally, substitute back to find the value of the other variable.

Solve the first equation for $ x $ in terms of $ y $.
Substitute the expression for $ x $ into the second equation to solve for $ y $.
Substitute the value of $ y $ back into the expression for $ x $ to find the value of $ x $.

Step 1: Solve for $ x $

From the first equation $ x - 4y = 21 $, we can express $ x $ in terms of $ y $: \[ x = 4y + 21 \]

Step 2: Substitute and Solve for $ y $

Next, we substitute $ x $ into the second equation $ 3x - 5y = 35 $: \[ 3(4y + 21) - 5y = 35 \] Expanding this gives: \[ 12y + 63 - 5y = 35 \] Combining like terms results in: \[ 7y + 63 = 35 \] Subtracting 63 from both sides yields: \[ 7y = -28 \] Dividing by 7 gives: \[ y = -4 \]

Step 3: Substitute $ y $ Back to Find $ x $

Now we substitute $ y = -4 $ back into the expression for $ x $: \[ x = 4(-4) + 21 = -16 + 21 = 5 \]