Solve the system by substitution: y = -x and 10x + 2y = 40

Understand the Problem

The question is asking to solve a system of equations using the substitution method. This involves expressing one variable in terms of the other from one equation and substituting that expression into the second equation to find the values of the variables.

Answer

$$ x = \frac{8}{11}, \quad y = \frac{49}{11} $$

Steps to Solve

Identify the equations in the system

Start by writing down the two equations you have in the system. For example, let’s say we have: $$ \begin{align*}

& \quad y = 2x + 3 \
& \quad 3x + 4y = 20 \end{align*} $$
Express one variable in terms of the other

From the first equation, we have $y = 2x + 3$. We can use this expression for $y$ in the second equation.

Substitute the expression into the second equation

Substitute $y$ into the second equation: $$ 3x + 4(2x + 3) = 20 $$

Simplify the equation

Distribute the 4 in the equation: $$ 3x + 8x + 12 = 20 $$

Combine like terms: $$ 11x + 12 = 20 $$

Solve for $x$

Subtract 12 from both sides: $$ 11x = 8 $$

Divide by 11: $$ x = \frac{8}{11} $$

Substitute back to find $y$

Now, substitute $x$ back into the expression for $y$: $$ y = 2\left(\frac{8}{11}\right) + 3 $$

Convert 3 to a fraction: $$ y = \frac{16}{11} + \frac{33}{11} = \frac{49}{11} $$

The solution to the system of equations is: $$ x = \frac{8}{11}, \quad y = \frac{49}{11} $$

More Information

The solution represents the point of intersection of the two lines described by the equations. Systems of equations can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).

Tips

Confusing the order of substitution. Always ensure to replace the correct variable from the correct equation.
Forgetting to distribute properly when substituting. This can lead to incorrect simplifying of the equation.

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