In Exercises 5–18, solve each system by the substitution method. 2x + 5y = - 4 3x - y = 11

Verified step by step guidance

1. Start with the given system of equations: $\mathrm{2x}+5y=-43x-y=11$.
2. Solve one of the equations for one variable in terms of the other. For example, solve the second equation for $y$: $\mathrm{3x}-y=\; 11$ becomes $y=\; 3x\; -\; 11$.
3. Substitute the expression for $y$ from step 2 into the first equation: replace $y$ in $2x\; +\; 5y\; =\; -4$ with $3x\; -\; 11$ to get $2x\; +\; 5(3x\; -\; 11)\; =\; -4$.
4. Simplify and solve the resulting equation for $x$. This will give you the value of $x$ in terms of numbers.
5. Substitute the value of $x$ back into the expression for $y$ found in step 2 to find the corresponding value of $y$. This completes the solution to the system.

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 inverse operations such as addition, subtraction, multiplication, or division. Mastery of these algebraic manipulations is essential to find the values of variables accurately.