Question

Solve the system by substitution.

$$\begin{gathered} x+3y=10 \\ 4x+y=18 \end{gathered}$$

Short answer

The solution set is $(4,2)$.

Step-by-step solution

Step 1. Given Information

We have to solve the system by substitution.

$$\begin{gathered} x+3y=10 \\ 4x+y=18 \end{gathered}$$

Step 2. Firstly solve the equation x+3y=10 for the variable src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMTkiIHdpZHRoPSIxMCIgd3JzOmJhc2VsaW5lPSIxNSI+PCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtaT54PC9taT48L21hdGg+LS0+PGRlZnM+PHN0eWxlIHR5cGU9InRleHQvY3NzIi8+PC9kZWZzPjx0ZXh0IGZvbnQtZmFtaWx5PSJBcmlhbCIgZm9udC1zaXplPSIxNiIgZm9udC1zdHlsZT0iaXRhbGljIiB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI0LjUiIHk9IjE1Ij54PC90ZXh0Pjwvc3ZnPg==" role="math" localid="1648476788711" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/d6a55f08-1b94-47a8-ab78-6e0c0ec2fe36.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220328%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220328T142146Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=eef95b6eaa6a4bab4bd2afa045a96380c78406f50d0a2f1988948ca1838f25a0" x.

$$\begin{gathered} \mathrm{Subtract}3y\mathrm{on}\mathrm{both}\mathrm{side} \\ x+3y-3y=10-3y \\ x=10-3y \end{gathered}$$

Step 3. Putting the value of x in the second equation.

$$\begin{gathered} 4(10-3y)+y=18 \\ 40-12y+y=18 \\ 40-11y=18 \\ \mathrm{Subtract}40\mathrm{on}\mathrm{both}\mathrm{side} \\ 40-11y-40=18-40 \\ -11y=-22 \\ \mathrm{Divide}\mathrm{by}-11\mathrm{on}\mathrm{both}\mathrm{side} \\ \frac{-11}{-11}y=\frac{-22}{-11} \\ y=2 \end{gathered}$$

Step 4. Now putting y=2 on the equation x=10-3y

$$\begin{gathered} x=10-3\times 2 \\ x=10-6 \\ x=4 \end{gathered}$$