To solve the given linear equations and find their parametric representations, let's begin with the first equation:
1. Find a parametric representation of the solution set of the linear equation:
$12x_1+24x_2-36x_3=12$
To find the parametric representation, we need to express the variables in terms of one or more parameters. We'll solve this equation step by step.
Step 1: Rewrite the equation in parametric form by expressing the variables as linear combinations of parameters. Let's introduce two parameters, say t and s, and express the variables as follows:
$x_1=t$
$x_2=s$
$x_3=???$
We still need to find the expression for $x_3$. To do that, we can solve the equation in terms of the parameters. Let's continue to the next step.
Step 2: Substitute the expressions for $x_1$ and $x_2$ into the original equation:
$12(t)+24(s)-36x_3=12$
Simplifying this equation gives us:
$12t+24s-36x_3=12$
Now, isolate $x_3$ to find its expression in terms of the parameters:
$-36x_3=12-12t-24s$
$x_3=\frac{12-12t-24s}{-36}$
$x_3=\frac{-1+t+2s}{3}$
Step 3: Now we have expressions for $x_1$, $x_2$, and $x_3$ in terms of the parameters t and s. Therefore, the parametric representation of the solution set for this equation is:
$x_1=t$
$x_2=s$
$x_3=\frac{-1+t+2s}{3}$
Moving on to the second equation:
2. Find a parametric representation of the solution set of the linear equation:
$\frac{3}{5}{x}_1+7x_2=15$
Again, we'll follow the same steps as before.
Step 1: Express the variables in terms of parameters:
$x_1=???$
$x_2=???$
Step 2: Substitute the expressions for $x_1$ and $x_2$ into the equation:
$\frac{3}{5}(x_1)+7(x_2)=15$
Simplifying this equation gives us:
$\frac{3}{5}{x}_1+7x_2=15$
To proceed, let's isolate one variable. In this case, let's solve for $x_1$:
$\frac{3}{5}{x}_1=15-7x_2$
$x_1=\frac{15-7x_2}{\frac{3}{5}}$
$x_1=\frac{5(15-7x_2)}{3}$
$x_1=\frac{75-35x_2}{3}$
Step 3: Now we can express $x_1$ and $x_2$ in terms of a parameter, say t:
$x_1=\frac{75-35t}{<br>3}$
$x_2=t$
Therefore, the parametric representation of the solution set for this equation is:
$x_1=\frac{75-35t}{3}$
$x_2=t$
In summary, we have found the parametric representations for both linear equations:
1. For the equation $12x_1+24x_2-36x_3=12$, the parametric representation is:
$x_1=t$
$x_2=s$
$x_3=\frac{-1+t+2s}{3}$
2. For the equation $\frac{3}{5}{x}_1+7x_2=15$, the parametric representation is:
$x_1=\frac{75-35t}{3}$
$x_2=t$
These representations allow us to express the solution sets of the equations in terms of parameters, providing a systematic way to explore their solutions.