Problem 150 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 11 x+9 y=-5 \\ 7 x+5 y=-1 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \((2, -3)\).

Step 1: Solve one of the equations for one variable

Choose one of the equations and solve it for one of the variables. For example, solve the second equation for y. \(7x + 5y = -1\)Solve for y: \(5y = -7x - 1\) \(y = -\frac{7}{5}x - \frac{1}{5}\)

Step 2: Substitute the expression for y into the first equation

Now take the expression found for \(y\) from Step 1 and substitute it into the first equation. First equation: \(11x + 9y = -5\) Replace \(y\) with \( -\frac{7}{5}x - \frac{1}{5}\): \(11x + 9(-\frac{7}{5}x - \frac{1}{5}) = -5\)

Step 3: Simplify and solve for x

Distribute and combine like terms to solve for \(x\): \(11x + 9(-\frac{7}{5}x - \frac{1}{5}) = -5 \) \(11x - \frac{63}{5}x - \frac{9}{5} = -5\) Multiply entire equation by 5 to eliminate fractions: \(5(11x) - 63x - 9 = -25\)\(55x - 63x - 9 = -25\) Combine like terms:\(-8x - 9 = -25\)Isolate \( x \): \(-8x = -16\)\(x = 2\)

Step 4: Substitute x back into the expression for y

Now that \(x\) is found, substitute \(x\) into the expression found for \(y\) in Step 1: \(y = -\frac{7}{5}(2) - \frac{1}{5}\) \(y = -\frac{14}{5} - \frac{1}{5} \) \(y = -3\)

Step 5: Write the solution as an ordered pair

Combine the values of \(x\) and \(y\) to write the solution as an ordered pair: \((2, -3)\)

Key Concepts

substitution method

Solving a system of equations by substitution involves finding one variable in terms of the other from one equation. Then, you substitute this expression into the other equation. This method is useful when one equation is already solved for one variable or can be easily manipulated to do so. This step-by-step approach helps in sequentially eliminating variables and making the problem less complex to solve.

linear equations

Linear equations are equations of the first degree. They graph as straight lines. They have no exponents higher than one and their solutions are straight-line graphs. For instance, in the system of equations given to us: \( 11x + 9y = -5 \) and \( 7x + 5y = -1 \), both equations represent straight lines on a coordinate plane. When solving such systems, you are essentially finding the point where these lines intersect.

algebraic expressions

Algebraic expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, and division). In our problem, both equations are algebraic expressions. To solve the system using substitution, you simplify these expressions by isolating a variable. In this case, we rearrange the equation \( 7x + 5y = -1 \) to solve for \( y \) and get \( y = -\frac{7}{5}x - \frac{1}{5} \). Substitution and further algebraic manipulations help in solving these equations.

ordered pairs

An ordered pair represents a point on a coordinate plane, given in the form \((x, y)\). Each solution of a system of equations corresponds to an ordered pair where the graphs of the equations intersect. In our solution, after finding \( x = 2 \) and \( y = -3 \), we write the solution as the ordered pair \((2, -3)\). This pair is the point where the lines represented by our equations meet, showing the values of \( x \) and \( y \) that satisfy both equations simultaneously.