Problem 27 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{array}{l} {-3 w+z=4} \\ {-9 w+5 z=-1} \end{array} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(w = -7/2\) and \(z = -13/2\).

Step 1: Isolate 'z' in the first equation

Multiply the first equation by 5 to ensure that coefficient of 'z' is the same in both equations: \[5 (-3w + z) = 5*4\] We now have the equation \[-15w + 5z = 20\]

Step 2: Substitute 'z' from the first equation into the second equation

We can express 'z' from the first equation as \[z = 15w + 20 / 5 = 3w + 4\] and substitute this to the second equation leading to \[-9w + 5(3w + 4) = -1\] We can simplify this equation to \[6w + 20 = -1\] And, then by subtracting 20 from both sides, we get \[6w = -21\] and by dividing by 6 we find the value of w, i.e. \[w = -21/6 = -7/2\]

Step 3: Substitute the value of 'w' into the first equation to solve for 'z'

Substitute 'w' into the first equation \[-3*(-7/2) + z = 4\] This simplifies down to \[21/2 + z = 4\] and subtracting 21/2 from both sides gives, z = 4 - 21/2 = -13/2

Key Concepts

Understanding Linear Systems

Linear systems are a set of two or more equations that represent straight lines when graphed on a coordinate plane. Each equation in the system will have variables, often referred to as unknowns, which need to be solved. The aim is to find the values of these variables that satisfy all the given equations simultaneously. For example, in a system of two equations: \(-3w + z = 4\) and \(-9w + 5z = -1\), we are looking for the exact pair of values \(w\) and \(z\) that make both these equations true at the same time. This often represents the point where the two lines intersect on the graph.

Detailed Solution Steps Using Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. Here is how it's done step-by-step:

• Step 1: Isolate a variable
  First, we need to isolate 'z' in the first equation. This makes it easier to substitute it into the second equation later on. We adjust the equation so that 'z' stands by itself on one side.
• Step 2: Substitute into the second equation
  Once 'z' is isolated, we rewrite the first equation in terms of 'z' and substitute this expression into the second equation. This step merges the two equations into one single equation that only has one variable - 'w'.
• Step 3: Solve for the isolated variable
  We solve the derived equation for 'w'. After finding the value of 'w', we go back and substitute it into the original equation to find the value of 'z'.

You follow these repeatable steps till you untangle the values of all unknowns.

Isolation of a Variable

Isolating a variable is a crucial step when solving linear systems using the substitution method. It simplifies one equation in terms of one variable. For the equation \(-3w + z = 4\), the focus is on getting 'z' by itself. We make adjustments through addition, subtraction, multiplication, or division to clear out coefficients and constants around 'z'. This results in something like \(z = 3w + 4\).

The goal is to then replace the isolated variable in another equation. By replacing 'z' from the first equation into the second one, we eliminate one of the variables and can solve for the other variable easily. Proper isolation of a variable can make the subsequent steps simpler by breaking down complex systems into single variable problems.