In Exercises 5–18, solve each system by the substitution method. 5x + 2y = 0 x - 3y = 0

Verified step by step guidance

1. Start with the given system of equations: $\mathrm{5x}+2y=0x-3y=0$.
2. Solve one of the equations for one variable in terms of the other. For example, from the second equation $x-3y=0$, solve for $x$: $x=3y$.
3. Substitute the expression for $x$ from step 2 into the first equation: replace $x$ with $3y$ in $5x\; +\; 2y\; =\; 0$ to get $5(3y)\; +\; 2y\; =\; 0$.
4. Simplify the resulting equation and solve for $y$: $15y\; +\; 2y\; =\; 0$ which simplifies to $17y\; =\; 0$, then solve for $y$.
5. Use the value of $y$ found in step 4 and substitute it back into the expression for $x$ from step 2 to find the value of $x$.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and represent these systems is fundamental to solving them.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve step-by-step.

Solving Linear Equations

Solving linear equations requires isolating the variable using algebraic operations such as addition, subtraction, multiplication, and division. Mastery of these operations is essential to find the exact values of variables in the system.