Questions: Consider the system 1/2 x+y=20; x+4 y=70. Which property allows you to write the system as -x-2 y=-40; x+4 y=70 ?

Transcript text: Consider the system $\left\{\begin{array}{l}\frac{1}{2} x+y=20 \\ x+4 y=70\end{array}\right.$ Which property allows you to write the system as $\left\{\begin{array}{l}-x-2 y=-40 \\ x+4 y=70\end{array}\right.$ ?

Solution

Solution Steps

To rewrite the system of equations, we can multiply the first equation by -2. This is an application of the property that allows us to multiply both sides of an equation by the same non-zero number to obtain an equivalent equation.

Step 1: Understanding the Original System of Equations

The original system of equations is given by: \[ \begin{cases} \frac{1}{2}x + y = 20 \\ x + 4y = 70 \end{cases} \]

Step 2: Applying the Multiplication Property

To transform the first equation, we multiply both sides by $-2$. This is based on the property that allows us to multiply both sides of an equation by the same non-zero number to obtain an equivalent equation.

Step 3: Transforming the First Equation

Multiplying the first equation by $-2$, we get: \[ -2 \left(\frac{1}{2}x + y\right) = -2 \times 20 \] Simplifying, this becomes: \[ -x - 2y = -40 \]

Step 4: Writing the Transformed System

The transformed system of equations is: \[ \begin{cases} -x - 2y = -40 \\ x + 4y = 70 \end{cases} \]