Question

How do you solve $y=6x$ and $2x+3y=-20$ using substitution? $$$$

Answer 1

Hint: We recall that from substitution method that if we are given two linear equation $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ then we express $y$ in terms of $x$ from one of the expression and put $y$ in terms of $x$ in other equation to get the value of $x$. We are already given the first equation $y=6x$ in terms of $x$. We put $y$ in the second equation and solve for $x$ and then put $x$ in first equation to get $y$.$$$$

Complete step-by-step answer:
We know that the general linear equation is given by $ax+by+c=0$ where $a,b,c$ are real numbers and $a\ne 0,b\ne 0$ .We need at least two equations to find a unique solution. We are given the following pair of equations in the equation in the question.
$$\begin{array}{rl}& y=6x.....\left(1\right)\\ & 2x+3y=-20....\left(2\right)\end{array}$$
We know from substation methods to solve linear equations that we have to express $y$ in terms of $x$ from one of the equations and then put $y$ in the other equation. We see that the equation (1) is already given to us $y$ as an expression of $x$. So we put $y=6x$ in equation (2) to have
$$\begin{array}{rl}& \Rightarrow 2x+3\left(6x\right)=-20\\ & \Rightarrow 2x+18x=-20\\ & \Rightarrow 20x=-20\end{array}$$
 We see the above expression is now a linear equation only in one variable that is $x$. We divide both sides of the above equation by 20 to have
$$\Rightarrow x=-1$$
We put obtained value of $x=-1$ in equation (1) to have
$$y=6(-1)=-6$$
So the solution of the given equations is $x=-1,y=-6$$$$$

Note: We can put $x=-1,y=6$ in the left hand side of second equation to have$2(-1)+3(-6)=-20$. Hence our solution is verified. We note that if $a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0$ be two linear equations then we can oblation unique solution only when ratio between coefficients of variables is not equal that is ${\displaystyle \frac{a_1}{a_2}}\ne {\displaystyle \frac{b_1}{b_2}}$. Here in this problem we have a ratio of coefficients ${\displaystyle \frac{6}{2}}\ne {\displaystyle \frac{-1}{3}}$.We should check this in rough before solving the equation.